Quantum Monte Carlo study of few-and many-body bose

Quantum Monte Carlo study of few-and many-body bose

systems in one and two dimensions

Grecia Guijarro Gámez

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DOCTORAL THESIS

QUANTUM MONTE CARLO STUDY OF FEW-

AND MANY-BODY BOSE SYSTEMS IN ONE

AND TWO DIMENSIONS

Author: Grecia Guijarro Gamez

Supervisors:

Prof. Dr. Grigory Astrakharchik

Prof. Dr. Jordi Boronat Medico

DOCTORAL PROGRAM IN COMPUTATIONAL AND APPLIED

PHYSICS

Barcelona 2020

ABSTRACT

In this Thesis, we report a detailed study of the ground-state properties of a

set of quantum few- and many-body systems by using Quantum Monte Carlo

methods. First, we introduced the Variational Monte Carlo and Diffusion Monte

Carlo methods, these are the methods used in this Thesis to obtain the proper-

ties of the systems. The first systems we studied consist of few-body clusters in

a one-dimensional Bose-Bose and Bose-Fermi mixtures. Each mixture is formed

by two different species with attractive interspecies and repulsive intraspecies

contact interactions. For each mixture, we focused on the study of the dimer,

tetramer, and hexamer clusters. We calculated their binding energies and unbind-

ing thresholds. Combining these results with a three-body theory, we extracted

the three-dimer scattering length close to the dimer-dimer zero crossing. For

both mixtures, the three-dimer interaction turns out to be repulsive. Our results

constitute a concrete proposal for obtaining a one-dimensional gas with a pure

three-body repulsion. The next system analyzed consists of few-body clusters

in a two-dimensional Bose-Bose mixture using two types of interactions. The

first case corresponds to a bilayer of dipoles aligned perpendicularly to the planes

and, in the second, we model the interactions by finite-range Gaussian poten-

tials. We find that all the considered clusters are bound states and that their

energies are universal functions of the scattering lengths, for sufficiently large

attraction-to-repulsion ratios. Studying the hexamer energy close to the corre-

sponding threshold, we discovered an effective three-dimer repulsion, which can

stabilize interesting many-body phases. Once the existence of the bound states

in the dipolar bilayer has been demonstrated, we investigated whether halos can

occur in this system. A halo state is a quantum bound state whose size is much

iii

iv

larger than the range of the attractive interaction between the atoms that form

it, showing universal ratios between energy and size. For clusters composed from

three up to six dipoles, we find two very distinct halo structures. For large in-

terlayer separation, the halo structure is roughly symmetric. However, for the

deepest bound clusters and as the clusters approach the threshold, we discover

an unusual shape of the halo states, highly anisotropic. Importantly, our results

prove the existence of stable halo states composed of up to six particles. To the

best of our knowledge, this is the first time that halo states with such a large

number of particles have been predicted and observed in a numerical simulation.

The next system we studied is a two-dimensional many-body dipolar fluid con-

fined to a bilayer geometry. We calculated the ground-state phase diagram as a

function of the density and the separation between layers. Our simulations show

that the system undergoes a phase transition from a gas to a stable liquid as the

interlayer distance increases. The liquid phase is stable in a wide range of den-

sities and interlayer values. In the final part of this Thesis, we studied a system

of dipolar bosons confined to a multilayer geometry formed by equally spaced

two-dimensional layers. We calculated the ground-state phase diagram as a func-

tion of the density, the separation between layers, and the number of layers. The

key result of our study in the dipolar multilayer is the existence of three phases:

atomic gas, solid, and gas of chains, in a wide range of the system parameters.

Remarkably, we find that the density of the solid phase decreases several orders of

magnitude as the number of layers in the system increases. The results reported

in this Thesis show that a dipolar system in a bilayer and multilayer geome-

tries offer stable and highly controllable setups for observing interesting phases

of quantum matter, such as halo states, and ultra-dilute liquids and solids.

RESUMEN

En esta Tesis, presentamos un estudio detallado de las propiedades del estado

fundamental de un conjunto de sistemas cuanticos de pocos y muchos cuerpos

mediante el uso de los metodos de Monte Carlo Cuantico. Primero, introducimos

los metodos de Monte Carlo Variacional y Monte Carlo Difusivo que usamos en

esta Tesis para obtener las propiedades de los sistemas. Los primeros sistemas que

estudiamos son cumulos de pocos cuerpos en mezclas unidimensionales de Bose-

Bose y Bose-Fermi. Cada una de las mezclas esta formada por dos especies con

interacciones atractivas para interespecies y repulsivas para intraespecies. Para

cada una de las mezclas nos enfocamos en el estudio de dımeros, tetrameros y

hexameros. Calculamos las energıas de ligadura y los valores umbrales de rup-

tura de los cumulos. Combinando estos resultados con una teorıa de tres cuerpos,

extraemos la longitud de dispersion de tres dımeros cerca del punto de ruptura

dımero-dımero. Para ambas mezclas la interaccion de tres dımeros resulta ser

repulsiva. El siguiente sistema analizado son cumulos de pocos cuerpos en una

mezcla bidimensional de Bose-Bose con dos tipos de interacciones. El primer

caso corresponde a una bicapa de dipolos con momentos dipolares orientados

perpendicularmente a los planos y, en el segundo, modelamos las interacciones

con potenciales gaussianos de rango finito. Encontramos que para relaciones de

atraccion-repulsion suficientemente grandes todos los cumulos considerados son

estados ligados y sus energıas son funciones universales de las longitudes de dis-

persion. Estudiando la energıa del hexamero cerca del punto umbral correspon-

diente, descubrimos una repulsion efectiva de tres dımeros, que puede estabilizar

fases interesantes de muchos cuerpos. Despues de demostrar la existencia de los

estados ligados en la bicapa dipolar, investigamos si pueden ocurrir estados de

v

vi

halo en este sistema. Un estado de halo es un estado ligado cuantico cuyo tamano

es mucho mayor que el rango de la interaccion atractiva entre los atomos que lo

forman. Para cumulos compuestos de tres hasta seis dipolos encontramos dos

estructuras de halo muy distintas. Para separaciones grandes entre las capas, la

estructura de halo es aproximadamente simetrica. Sin embargo, para los estados

mas ligados y a medida que los cumulos se acercan al punto umbral, descubri-

mos una estructura de halo inusual, altamente anisotropica. Nuestros resultados

demuestran la existencia de estados de halo estables compuestos de hasta seis

dipolos. Hasta donde sabemos, esta es la primera vez que estados de halo con

un numero tan grande de partıculas se predicen y observan en una simulacion

numerica. El siguiente sistema estudiado es un fluido bidimensional dipolar de

muchos cuerpos confinado a una geometrıa de bicapa. Calculamos el diagrama

de fases del estado fundamental como funcion de la densidad y de la separacion

entre las capas. Nuestras simulaciones muestran que en el sistema ocurre una

transicion de fase, de un gas a un lıquido a medida que se incrementa la distancia

entre las capas. El lıquido es estable en una region amplia de densidades y de la

distancia entre las capas. En la parte final de esta Tesis se estudia un sistema

de bosones dipolares confinados a una geometrıa multicapa formada por capas

bidimensionales igualmente espaciadas. Calculamos el diagrama de fases del es-

tado fundamental como funcion de la densidad, la separacion entre las capas y

el numero de capas. El resultado clave de nuestro estudio sobre la multicapa es

la existencia de tres fases: gas atomico, solido y gas de cadenas, en una region

amplia de los parametros del sistema. Encontramos que la densidad del solido

disminuye varios ordenes de magnitud a medida que el numero de capas en el sis-

tema aumenta. Los resultados reportados en esta Tesis muestran que un sistema

de dipolos confinados a una bicapa o multicapa ofrecen configuraciones estables

y altamente controlables para observar fases interesantes de materia cuantica.

CONTENTS

Abstract ii

Resumen v

1 Introduction 1

2 Quantum Monte Carlo Methods 7

2.1 The Quantum Many-Body Problem . . . . . . . . . . . . . . . . . 8

2.2 Variational Monte Carlo Method . . . . . . . . . . . . . . . . . . 9

2.2.1 Variational Principle . . . . . . . . . . . . . . . . . . . . . 9

2.2.2 The Method . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.3 Stochastic Processes . . . . . . . . . . . . . . . . . . . . . 11

2.2.4 Markov Processes . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.5 Metropolis Algorithm . . . . . . . . . . . . . . . . . . . . . 13

2.2.6 VMC Stochastic Realization . . . . . . . . . . . . . . . . . 14

2.3 Diffusion Monte Carlo Method . . . . . . . . . . . . . . . . . . . . 15

2.3.1 Green’s Function . . . . . . . . . . . . . . . . . . . . . . . 17

2.3.2 Importance Sampling . . . . . . . . . . . . . . . . . . . . . 18

2.3.3 Importance-Sampling Green’s Function and Short-Time Ap-

proximation . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3.4 DMC Stochastic Realization . . . . . . . . . . . . . . . . . 22

2.3.5 Convergence Analysis . . . . . . . . . . . . . . . . . . . . . 24

2.4 Trial Wave Functions . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.5 Quantum Monte Carlo Estimators . . . . . . . . . . . . . . . . . . 28

vii

viii CONTENTS

2.5.1 Pair Distribution Function . . . . . . . . . . . . . . . . . . 28

2.5.2 One-Body Density Matrix . . . . . . . . . . . . . . . . . . 29

2.5.3 Mixed Estimators and Extrapolation Technique . . . . . . 29

2.5.4 Pure Estimators . . . . . . . . . . . . . . . . . . . . . . . . 30

3 One-Dimensional Three-Boson Problem with Two- and Three-

Body Interactions 33

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2 The System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.3 Details of the Methods . . . . . . . . . . . . . . . . . . . . . . . . 36

3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.4.1 Binding Energies . . . . . . . . . . . . . . . . . . . . . . . 38

3.4.2 Threshold Determination . . . . . . . . . . . . . . . . . . . 38

3.4.3 Three-Dimer Repulsion . . . . . . . . . . . . . . . . . . . . 39

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4 Few-Body Bound States of Two-Dimensional Bosons 43

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.2 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . 45


4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.4.1 Binding Energies . . . . . . . . . . . . . . . . . . . . . . . 48

4.4.2 Threshold Determination . . . . . . . . . . . . . . . . . . . 51

4.4.3 Three-Dimer Repulsion . . . . . . . . . . . . . . . . . . . . 53

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5 Quantum Halo States in Two-Dimensional Dipolar Clusters 59

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.2 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . 62


5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.4.1 Structure of the Bound States . . . . . . . . . . . . . . . . 63

5.4.2 Quantum Halo Characteristics . . . . . . . . . . . . . . . . 67

5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

CONTENTS ix

6 Quantum Liquid of Two-Dimensional Dipolar Bosons 73

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6.2 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . 75


6.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.4.1 Finite Size Effects . . . . . . . . . . . . . . . . . . . . . . . 77

6.4.2 Equation of State . . . . . . . . . . . . . . . . . . . . . . . 79

6.4.3 Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . 80

6.4.4 Depletion of the Condensate . . . . . . . . . . . . . . . . . 82

6.4.5 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

7 Phases of Dipolar Bosons Confined to a Multilayer Geometry 89

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

7.2 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . 90


7.3.1 Gas Trial Wave Function . . . . . . . . . . . . . . . . . . . 92

7.3.2 Solid Trial Wave Function . . . . . . . . . . . . . . . . . . 93

7.3.3 Gas of Chains Trial Wave Function . . . . . . . . . . . . . 94

7.4 Details of the Gas of Chains Wave Function . . . . . . . . . . . . 95

7.4.1 Construction of Trial Wave Function . . . . . . . . . . . . 96

7.4.2 Many-Body Term: Gaussian Function . . . . . . . . . . . . 97

7.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

7.5.1 Crystallization and Threshold Densities . . . . . . . . . . . 99

7.5.2 Dipoles within a Four-Layer Geometry . . . . . . . . . . . 102

7.5.3 Dipoles within an M-Layer Geometry . . . . . . . . . . . . 106

7.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

8 Conclusions 109

A Trial Wave Function 113

B Symmetric Trial Wave Function 117

Bibliography 122

Agradecimientos 138

CHAPTER 1

INTRODUCTION

In this Thesis, we report a detailed study of the ground-state properties of a

set of quantum few- and many-body systems in one and two dimensions with

different types of interactions. Nevertheless, the main focus of this work is the

study of the ground-state properties of an ultracold Bose system with dipole-

dipole interaction between the particles. We consider the cases where the bosons

are confined to a bilayer and multilayer geometries, that consist of equally spaced

two-dimensional layers. These layers can be experimentally realized by imposing

tight confinement in one direction. We specifically address the study of new

quantum phases, their properties, and transitions between them. One expects

these systems to have a rich collection of few- and many-body phases because

the dipole-dipole interaction is anisotropic and quasi long-range. We will now

present a short historical review of the experiments and theoretical predictions

that motivated the study of ultracold dipolar bosonic gases.

The Bose-Einstein condensation (BEC) is a quantum phenomenon occurring

when a macroscopic number of bosons occupy the zero momentum state. This

happens when the system reaches a temperature below a critical value. Although

BEC was predicted by Albert Einstein in 1924 [1] based on a previous work by

Satyendranath Bose, it was not until 1995 that this phenomenon was experimen-

tally observed in rubidium [2] and sodium [3] gases independently. The Nobel

Prize in Physics 2001 was awarded to Wolfgang Ketterle, Eric A. Cornell, and

Carl E. Wieman for the achievement of BEC in dilute gases of alkali atoms [4].

Typically the BEC state is reached for temperatures and densities of the order of

1

2 Chapter 1. Introduction

T ∼ 10−7K and n ∼ 1013− 1014cm−3, respectively. Since the experimental obser-

vation of the BEC, there have been intense theoretical and experimental efforts

to understand ultracold bosonic and fermionic gases. Interesting quantum phases

have been predicted and experimentally realized in these systems, for example,

quantum droplets in a mixture of Bose-Einstein condensates [5–8] and in dipo-

lar bosonic gases [9–12], quantum droplets in optical lattices [13, 14], impurities

of atoms immersed in a gas of fermions [15–17] or bosons [18–20], the so-called

polaron problem, among others. At very low densities, some ultracold gases can

be characterized by the s-wave scattering length, which means that they can be

described by an isotropic, short-range, contact interaction model. However, there

are gases with more complex interactions like dipolar interactions.

Recent experiments have enabled the experimental study of ultracold gases

with dipole-dipole interaction (DDI). The DDI has two main properties that

greatly distinguish it from the contact interactions. Firstly, DDI is long-range

in three dimensions, it falls off with a power-law 1/r3 dependence, where r is

the distance between particles. Secondly, DDI is anisotropic which means that

the interaction strength and its sign (repulsive or attractive), depends on the an-

gle between the polarization direction and the relative distance of the particles.

DDI can be found in magnetic atoms, ground-state heteronuclear molecules, and

Rydberg atoms, among others [21]. The first Bose-Einstein condensate of mag-

netic atoms was realized in a gas of chromium atoms in 2005 [22, 23]. The most

recent experiments of dipolar gases are done with Dysprosium [24, 25] and Er-

bium [26]. Many interesting phenomena have been observed and predicted in

dipolar gases, for example, dipolar Bose supersolid stripes [27], dipolar quantum

mixtures [28, 29], formation of a crystal phase [30, 31], and a pair superfluid [32–

34].

Describing a quantum many-body system is a demanding task, as it involves

the interactions of a large number of particles subject to spatial constraints. Only

for systems with very simple interactions, and under some assumptions, can the

Schrodinger equation be solved exactly. As we are studying systems with dipolar

interactions, complementary numerical methods become necessary, like in our

case, quantum Monte Carlo methods.

Quantum Monte Carlo (QMC) methods are a set of stochastic techniques that

are used to calculate the ground-state properties of quantum many-body systems

at zero or finite temperature [35–37]. One of the most used QMC techniques for

3

its simplicity is the Variational Monte Carlo (VMC) method. The VMC tech-

nique uses the variational principle of quantum mechanics to provide an upper

bound to the ground-state energy of a quantum system. The accuracy of this

method depends entirely on the accuracy of the trial wave function used to cal-

culate the expectation value of the Hamiltonian. Another QMC technique is the

Diffusion Monte Carlo (DMC) method that solves the many-body Schrodinger

equation in imaginary time. This method consists of evolving in imaginary time

the wave function of a quantum system, and after enough time has passed, it

projects out the ground state. The DMC method allows one to calculate the ex-

act ground-state energy of the system, as well as other properties, within control-

lable statistical errors. Both VMC and DMC methods have been shown to give an

accurate description of correlated quantum systems [37]. Examples include ultra-

dilute bosonic [38, 39] and fermionic mixtures [29], Bose [20, 40] and Fermi [17]

polarons, dipolar Bose supersolid stripes [27], Bose gas subject to a multi-rod

lattice [41], and ultracold quantum gases with spin-orbit interactions [42].

In this Thesis, we have used QMC methods to study the ground-state proper-

ties of a set of quantum few- and many-body systems. A large part of this Thesis

is focused on the study of dipolar Bose systems confined to a two-dimensional

bilayer and multilayer geometries. This Thesis is organized in the following way:

In Chapter 2, we explain the basics of the Quantum Monte Carlo methods

used in this Thesis. First, we present the Variational Monte Carlo method, which

is used to calculate an approximation to the ground-state energy of a quantum

system. Then, we introduce the Metropolis algorithm, a method used to generate

random numbers from an arbitrary probability distribution function. Afterwards,

we discuss the Diffusion Monte Carlo method, which allows one to calculate the

exact ground-state energy of bosonic systems at zero temperature. Later, we

describe a number of trial wave functions used for QMC calculations. Finally,

we show how several ground-state properties are evaluated in the Monte Carlo

algorithm.

In Chapter 3, we use the DMC method to calculate the ground-state prop-

erties of a one-dimensional Bose-Bose and Bose-Fermi mixtures with attractive

interspecies and repulsive intraspecies interactions. We focus on the study of the

tetramer and hexamer clusters. First, we describe the trial wave functions for

the system and the boundary conditions to be satisfied. Then, we evaluate the

tetramer and hexamer ground-state energies for Bose-Bose and Bose-Fermi mix-


tures. Afterwards, we determine the threshold for unbinding for the tetramer and

hexamer, where the clusters break into two and three dimers, respectively. Then,

combining these results with a one-dimensional three-body theory, we extract the

three-dimer scattering length close to the dimer-dimer zero crossing. Finally, we

discuss a mixture of ultracold gases for obtaining a one-dimensional gas with a

pure three-body repulsion.

In Chapter 4, we study the ground-state properties of few-body bound states

in a two-dimensional mixture of A and B bosons with two types of interactions.

The first case corresponds to a bilayer of dipoles and, in the second, we model

the interactions by non-local (separable) finite-range Gaussian potentials. First,

we show the details of the numerical techniques used to study the two models.

In the dipolar case, we use the diffusion Monte Carlo and in the Gaussian model,

we use the stochastic variational method. Then, using these methods we evaluate

the ground-state binding energies of the clusters. Also, we numerically determine

the threshold for unbinding of the bound states in the bilayer geometry. After-

wards, studying the hexamer energy near to the tetramer threshold allows us to

characterize an effective three-dimer interaction, which may have important im-

plications for the many-body problem, particularly for observing liquid states of

dipolar dimers in the bilayer geometry. Finally, we give some examples of dipolar

molecules as promising candidates for observing the predicted few-body states

within a bilayer setup.

In Chapter 5, we analyze the ground-state properties of loosely bound dipo-

lar states confined to a two-dimensional bilayer geometry by using the VMC

and DMC methods. We study dipolar dimers, trimers, tetramers, pentamers,

and hexamers. First, we evaluate the pair distribution functions for the dimer,

trimer, and tetramer for different values of the interlayer separation. Then, we

calculate the spatial distributions functions for the trimer and tetramer for two

characteristic interlayer distances. Knowledge of these structural properties per-

mits us to understand how the size and shape of the clusters change with the

interlayer distance. Finally, the calculations of the binding energies and sizes of

the clusters allow us to investigate whether quantum halos, bound states with a

wave function that extends deeply into the classically forbidden region, can occur

in this system.

In Chapter 6, we study a many-body system of dipolar bosons within a bilayer

geometry by using exact many-body quantum Monte Carlo methods. We consider

5

the case in which the dipoles are aligned perpendicularly to the parallel layers.

First, we describe the trial wave functions for the system. Then, we calculate the

equation of state (energy per particle as a function of the density) for different

values of the interlayer distance. Knowledge of the equation of state permits us

to establish the quantum phases present in the bilayer of dipoles. Afterwards,

we obtain the gas-liquid phase diagram of the dipolar fluid as a function of the

density and the separation between layers. Finally, we show numerical results for

the one-body density matrix, condensate fraction and polarization.

In Chapter 7, we use the diffusion Monte Carlo approach to study the ground-

state phase diagram of dipolar bosons in a geometry formed by equally spaced

two-dimensional layers. First, we discuss the trial wave functions to describe the

gas, solid, and gas of chains phases. In particular, for the trial function of the

chains, we have derived the expressions of the drift force and the local energy,

which are necessary to implement the DMC algorithm. Then, we consider the

case where there are four layers and the same number of dipoles in each layer.

In this case, we calculate the pair distribution functions for the different phases

present in the system. Also, we calculate the ground-state phase diagram as a

function of the total density and the interlayer distance. Finally, we consider the

case where the dipoles are confined to three up to ten layers. Here, we calculate

the zero-temperature phase diagram.

In Chapter 8, we present a summary of the principal results obtained in this

Thesis and the main conclusions achieved.

Pubications

The results of this doctoral research were published in:

• G. Guijarro, A. Pricoupenko, G. E. Astrakharchik, J. Boronat, and D. S.

Petrov, One-dimensional three-boson problem with two- and three-body

interactions, Phys. Rev. A 97, 061605(R) (2018).

• G. Guijarro, G. E. Astrakharchik, J. Boronat, B. Bazak, and D. S. Petrov,

Few-body bound states of two-dimensional bosons, Phys. Rev. A 101,

041602(R) (2020).

Manuscripts in process

• G. Guijarro, G. E. Astrakharchik, and J. Boronat, Quantum halo states in

two-dimensional dipolar clusters. Manuscript submitted for publication.


• G. Guijarro, G. E. Astrakharchik, and J. Boronat, Quantum liquid of two-

dimensional dipolar bosons. Manuscript in preparation.

• G. Guijarro, G. E. Astrakharchik, and J. Boronat, Phases of dipolar bosons

confined to a multilayer geometry. Manuscript in preparation.

CHAPTER 2

QUANTUM MONTE CARLO METHODS

The term Quantum Monte Carlo (QMC) refers to a set of stochastic techniques

whose objective is to solve as exactly as possible quantum many-body problems,

by determining the expectation values of quantum observables [35]. The QMC

methods have been demonstrated to give an accurate description of correlated

quantum systems at zero and low temperature [37]. Examples include ultracold

gases with bosonic [38, 39] and fermionic statistics [17, 29], quantum solids [43,

44], and Helium [45, 46].

To study systems at zero temperature, one can use the Variational Monte

Carlo method (VMC) or the Diffusion Monte Carlo method (DMC). The VMC

algorithm was introduced by McMillan in 1965 to study liquid Helium [47]. In

contrast, the DMC technique was developed in several works over the years [48].

The VMC method uses the variational principle of quantum mechanics to provide

an upper bound to the ground-state energy of a quantum system. On the other

hand, the DMC method allows one to calculate the exact ground-state energy,

of bosonic systems by solving the many-body Schrodinger equation in imaginary

time.

For fermionic systems, the DMC method provides an upper bound to the

ground-state energy and not the exact one [49]. This is because the wave function

of fermions is antisymmetric under the exchange of two particles. Therefore, there

are regions where it is positive and other regions where it is negative. This leads

to the so-called fermion sign problem.

To study quantum many-body systems with finite, but low temperature there

7

8 Chapter 2. Quantum Monte Carlo Methods

exists the Path Integral Monte Carlo (PIMC) method. This method is based on

the thermal density matrix and Feynman’s path-integral formulation of quantum

mechanics [50, 51].

In this chapter, we introduce the fundamental concepts of the Variational

Monte Carlo (VMC) and the Diffusion Monte Carlo (DMC) methods, which are

the Quantum Monte Carlo methods used in this Thesis. First, we discuss the

theoretical basis of the VMC method and its algorithm. Second, we present the

DMC method and its stochastic realization. Third, we discuss different types of

trial wave functions used in the method. Finally, we show how several ground-

state properties are evaluated in the Monte Carlo algorithm.

2.1 The Quantum Many-Body Problem

We will consider the generic quantum many-body problem involving N inter-

acting particles of mass m. We restrict ourselves to the case of particles in an

external potential Vext(ri) and pairwise interactions Vint(ri − rj). We can write

the Hamiltonian of such problems as

H = − ~2

2m

N∑i=1

∇2ri

+N∑i=1

Vext(ri) +N∑i=1

N∑j=i+1

Vint(ri − rj), (2.1)

where ri is the position of a single particle. It is difficult, if not impossible, to

exactly solve the Schrodinger equation for the many-body Hamiltonian, which in-

volves obtaining all its eigenstates. As the complete analytical solution is unavail-

able, we use numerical methods to calculate the wave function and the properties

of the ground state. We would like to calculate the ground-state expectation

value of an observable O

〈O〉 =〈Φ0|O|Φ0〉〈Φ0|Φ0〉

, (2.2)

Φ0 being the ground-state wave function. In particular, we are interested in

obtaining the ground-state energy of the system, which is defined as

E0 = 〈H〉 =〈Φ0|H|Φ0〉〈Φ0|Φ0〉

. (2.3)

Using Monte Carlo methods we can calculate the exact value of the ground-state

energy of a Bose system at zero temperature, within some statistical errors.

2.2. Variational Monte Carlo Method 9

2.2 Variational Monte Carlo Method

2.2.1 Variational Principle

The Variational Monte Carlo (VMC) method can be used to obtain an approxi-

mated value of the ground-state energy of a quantum system by using the varia-

tional principle of quantum mechanics. The variational principle states that the

expectation value of a Hamiltonian, H, obtained with a trial wave function |ΨT〉,provides an upper bound to the ground-state energy E0 of the system:

〈ΨT|H|ΨT〉〈ΨT|ΨT〉

≥ E0, (2.4)

if |ΨT〉 is not orthogonal to the ground-state wave function. The equality in

Eq. (2.4) is fulfilled only when the trial function |ΨT〉 is the exact ground-state

wave function. The proof of Eq. (2.4) is as follows. If |φn〉 is an eigenfunction

with eigenvalue En of H, the following properties are fulfilled

H|φn〉 = En|φn〉, 〈φn|φm〉 = δn,m, and∑n

|φn〉〈φn| = 1. (2.5)

Using these relations the expectation value of H can be written as

〈Ψ|H|Ψ〉〈Ψ|Ψ〉 =

∑n,m〈Ψ|φn〉〈φn|H|φm〉〈φm|Ψ〉∑

n〈Ψ|φn〉〈φn|Ψ〉

=

∑nEn〈φn|Ψ〉|2∑n |〈φn|Ψ〉|2

.

(2.6)

Since En ≥ E0, it follows that

〈Ψ|H|Ψ〉〈Ψ|Ψ〉 =

∑nEn|〈φn|Ψ〉|2∑n |〈φn|Ψ〉|2

≥∑

nE0|〈φn|Ψ〉|2∑n |〈φn|Ψ〉|2

= E0

∑n |〈φn|Ψ〉|2∑n |〈φn|Ψ〉|2

= E0,

(2.7)

and this proves the upper bound reported in Eq. (2.4). In general, the trial wave

function |ΨT〉 depends on a set of parameters that can be optimized in order to

find the lowest possible value of the energy. The trial wave function with these


optimal parameters is an approximation to the ground-state wave function of H

and the lowest energy is an upper bound to the ground-state energy.

2.2.2 The Method

In the Variational Monte Carlo (VMC) method one defines a normalized proba-

bility density function ρ(R)

ρ(R) =|ΨT(R)|2∫dR|ΨT(R)|2 , (2.8)

and a local energy EL(R)

EL(R) =1

ΨT(R)HΨT(R), (2.9)

here R = (~r1, · · · ,~rN) is a 3N -dimensional vector specifying the positions of N

particles. The expectation value of H can be written in the integral form

Evar =〈ΨT|H|ΨT〉〈ΨT|ΨT〉

=

∫dRΨ∗T(R)HΨT(R)∫

dR|ΨT(R)|2 =

∫dRρ(R)EL(R). (2.10)

The estimator of the variational energy Evar is then calculated as the mean value

of EL(R):

Evar =1

M

M∑k=1

EL(Rk), (2.11)

where M is the number of points Rk sampled from the probability density func-

tion ρ(R). As we mentioned before, ΨT depends on a set of parameters that are

optimized to minimize the energy. Therefore, we calculate the variational energy

Eq. (2.11) for several values of the parameters and obtain the minimum.

Other observables can also be calculated in the VMC method. The variational

expectation value of an observable O is given by

〈O〉var =

∫dRΨ∗T(R)OΨT(R)∫

dR|ΨT(R)|2 , (2.12)

which can be written as

〈O〉var =

∫dRρ(R)OL(R), (2.13)


where OL(R) is the local observable

OL(R) =1

ΨT(R)OΨT(R). (2.14)

The variational estimator of any local observable can be computed by averaging

the corresponding local value

〈O〉var =1

M

M∑k=1

OL(Rk). (2.15)

In general, the probability density ρ(R) = |ΨT(R)|2/∫dR|ΨT(R)|2 Eq. (2.8)

is complicated and depends on many variables, thus it cannot be sampled by

using other methods such as the rejection method [48]. The solution to this

problem is found in the Metropolis algorithm which will be discussed below. This

method is used to generate random numbers from any probability distribution by

constructing a Markov process. Before presenting the Metropolis algorithm, we

are going to introduce the concepts of stochastic processes and Markov processes.

2.2.3 Stochastic Processes

A stochastic process describes a time-dependent random variable R(t). For times

t1, t2, . . . , tn there exist a probability distribution P (R1, t1; R2, t2; . . . ; Rn, tn) where

R1, . . . ,Rn are random variables associated to R(t). Usually the times are or-

dered, t1 ≤ t2 ≤ . . . ≤ tn. We can write the probability distribution in terms of

the conditional probabilities as

P (Rn, tn; . . . ; R2, t2; R1, t1) =P (Rn, tn|Rn−1, tn−1; . . . ; R1, t1) . . .

× P (R2, t2|R1, t1)P (R1, t1).(2.16)

It is therefore clear that Rj is conditioned to Rj−1, . . . ,R1. To calculate the

probability distribution of a particular realization of R1, . . . ,Rn we need to do it

in order, this means, first calculate P (R1, t1) then P (R2, t2|R1, t1) and so on.


2.2.4 Markov Processes

A Markov process is a stochastic process for which the conditional probability for

the transition to a new state Rj depends only on the previous state Rj−1

P (Rj, tj|Rj−1, tj−1; . . . ; R1, t1) = P (Rj, tj|Rj−1, tj−1). (2.17)

Therefore for a Markov process we can rewrite Eq. (2.16) as

P (Rn, tn; . . . ; R2, t2; R1, t1) =P (Rn, tn|Rn−1, tn−1) . . .

× P (R2, t2|R1, t1)P (R1, t1).(2.18)

From here onwards we will consider Markov processes independent of time which

are known as stationary Markov processes. The probability P (Rf , |Ri) is called

the transition probability (or matrix) of going from an initial state Ri to a final

state Rf . The transition probability satisfy the following properties

P (Rf |Ri) ≥ 0, (2.19)∫dRfP (Rf |Ri) = 1. (2.20)

The last property simply means that given an initial state Ri, a posterior state

(the same or different) will be reached with certainty. Also, there is not a fully

absorbing state where the random walk stops.

We want to construct a Markov process that converges to the target proba-

bility distribution ρ(R) Eq. (2.8) by repeated applications of the transition prob-

ability. In order for this to happen several conditions must be met. The first one

is that the distribution ρ(R) must be an eigenvector of P (Rf |Ri) with eigenvalue

1 [36]∫dRiP (Rf |Ri)ρ(Ri) = ρ(Rf ) =

∫dRiP (Ri|Rf )ρ(Rf ) ∀Rf , (2.21)

this condition is known as stationarity condition, which means that is we start

from the target distribution ρ(R), after repeted applications of the transition

probability, we will continue to sample the target distribution ρ(R). In general it

is required that starting from any initial distribution ρini(R), it should converge

to the target distribution ρ(R) after applying the transition probability a finite


number of times,

limn→∞

∫dR1dR2 . . . dRnP (R|Rn)P (Rn|Rn−1) . . . P (R2|R1)ρini(R1)

= ρ(Rf ).

(2.22)

To ensure the convergence to a unique stationary distribution ρ(R) the Markov

process must be ergodic, which means that it must be possible to move between

any pair of states Rj and Rl in a finite number of steps, then all the states can

be visited. Another condition that the Markov process must fulfill is the detailed

balanced contition

P (Rf |Ri)ρ(Ri) = P (Ri|Rf )ρ(Rf ), (2.23)

for any states Ri and Rf . This condition imposes that the probability flux

between the states Ri and Rf to be the same in both directions.

2.2.5 Metropolis Algorithm

The Metropolis algorithm consists of a Markov process plus a decision criterium

on the random outcomes. We start with an initial state Ri. Then, we propose a

temporary state R′f according to a probability distribution Pprop(Rf |Ri), which

is known a priori. After that, we test the temporary state. If the temporary state

passes the test then we accept it as the new initial state. If it does not pass the

test then the initial state remains unchanged. The test consists of accepting the

move (the temporary state) with probability Pacc(Rf |Ri) or rejecting the move

with probability 1−Pacc(Rf |Ri). Notice that, the transition probability is given

by

P (Rf |Ri) =

Pacc(Rf |Ri)Pprop(Rf |Ri) if Rf 6= Ri

1−∫dR′fPacc(R

′f |Ri)Pprop(R′f |Ri) if Rf = Ri

(2.24)

where Pacc(Rf |Ri) is the probability of accepting the move. We are free to choose

the criterium for accepting a move, this means we are free to choose Pacc(Rf |Ri).

However, Pacc(Rf |Ri) has to fulfill the detailed balanced condition Eq. (2.23)

Pacc(Rf |Ri)

Pacc(Ri|Rf )=Pprop(Ri|Rf )ρ(Rf )

Pprop(Rf |Ri)ρ(Ri). (2.25)


The Metropolis algorithm makes a particular choice of Pacc(Rf |Ri)

Pacc(Rf |Ri) = min

{1,Pprop(Ri|Rf )ρ(Rf )

Pprop(Rf |Ri)ρ(Ri)

}. (2.26)

An advantage of this choice is that we do not need to calculate the normalization

factor for ρ(R), because it will cancel out.

To implement the Metropolis algorithm we need to choose a proposal proba-

bility Pprop(Rf |Ri). A simple choice of Pprop(Rf |Ri) is a normal Gaussian distri-

bution.

The Metropolis algorithm reads as:

1. Start from a random state Ri.

2. Propose a trial state R′ according to

R′ = Ri + χ,

where χ is an N-dimensional random vector sampled from a Gaussian dis-

tribution.

3. Calculate the quotient ρ(R′)/ρ(Ri).

4. Generate a random number ξ from the uniform distribution in [0, 1).

5. If ρ(R′)/ρ(Ri) > ξ the move is accepted and Ri+1 = R′. Otherwise stay in

the same state Ri+1 = Ri.

After applying the Metropolis algorithm a large enough number of times, the

Markov process will sampled the target distribution ρ(R).

Notice that in step 3 only the quotient ρ(R′)/ρ(Ri) defines the acceptance

probabilty because Pprop(Ri|R′) = Pprop(R′|Ri), since the Gaussian probability

distribution is symmetric.

2.2.6 VMC Stochastic Realization

Here we present the VMC algorithm:

1. We start with a random point R1 that represents the initial distribution

ρini(R) given by

ρini(R) = δ(R−R1).

2.3. Diffusion Monte Carlo Method 15

2. Using the Metropolis algorithm we construct the Markov process given by

{R1,R2, . . . ,RB, . . . ,RB+M}.

3. We remove the first B elements of the Markov process. The remaining

elements {R1,R2, . . . ,RM} (with the corresponding relabeling) are sampled

from the target distribution ρ(R).

4. Now we can calculate the variational estimator of the Hamiltonian

Evar =1

M

M∑k=1

EL(Rk). (2.27)

2.3 Diffusion Monte Carlo Method

In the VMC method, the accuracy of the energy Eq. (2.27) depends entirely

on the accuracy of the trial wave function. The larger the overlap between the

trial wave function and the ground-state wave function the better the estimation

of the ground-state energy. To overcome the limitations of the VMC method,

we introduce the Diffusion Monte Carlo (DMC) method. This method provides

a practical way of evolving in imaginary time the wave function of a quantum

system and obtaining, ultimately, the ground-state energy [52].

The starting point of the DMC method is the time-dependent many-body

Schrodinger equation with an energy shift ET, which is equivalent to replacing

H → H − ET

i~∂Ψ(R, t)

∂t=[H − ET

]Ψ(R, t)

=

[− ~2

2m∇2

R + V (R)− ET

]Ψ(R, t),

(2.28)

where R = (~r1, · · · ,~rN) is a 3N -dimensional vector specifying the coordinates

of all N particles, Ψ(R, t) is the many-body wave function of the system, which

depends on the particle coordinates and the time, and H is the many-body Hamil-

tonian Eq. (2.1)

∇2R =

N∑i=1

∇2ri, V (R) =

N∑i=1

Vext(ri) +N∑i=1

N∑j=i+1

Vint(ri − rj). (2.29)

Let us now perform a transformation from real time to imaginary time by intro-


ducing the new variable τ = it/~. After this, the Schrodinger equation Eq. (2.28)

becomes

−∂Ψ(R, τ)

∂τ=[H − ET

]Ψ(R, τ)

=[−D∇2

R + V (R)− ET

]Ψ(R, τ),

(2.30)

where D = ~2/2m. Eq. (2.30) can be identified as a modified diffusion equation

in the 3N - dimensional space. If the [V (R)− ET] term were removed, Eq. (2.30)

becomes the usual diffusion equation with a difussion constant D. On the other

hand, if the term with the Laplacian were removed, Eq. (2.30) would be a rate

equation, describing and exponential growth or decrease of the function Ψ(R, τ).

The objective is to solve Eq. (2.30) to access the ground state of the system.

Using the spectral descomposition

e−(H−ET)τ =∑i

|Φi〉e−(Ei−ET)τ 〈Φi|, (2.31)

the formal solution of Eq. (2.30)

|Ψ(R, τ)〉 = e−(H−ET)τ |Ψ(R, 0)〉, (2.32)

can be expressed as

|Ψ(R, τ)〉 =∑i=0

e−(Ei−ET)τ |Φi〉〈Φi|Ψ(R, 0)〉, (2.33)

where {Φi} and {Ei}, with H|Φi〉 = Ei|Φi〉, denote a complete sets of eigenfunc-

tions and eigenvalues of H, respectively. We consider that the eigenvalues are

ordered

E0 < E1 ≤ E2 ≤ . . . (2.34)

The amplitudes of each one of the terms in Eq. (2.33) can increase or decrease in

time depending on the sign of (En −ET). Notice that, for sufficiently long times

τ → ∞ the operator e−(H−ET)τ projects out the lowest eigenstate |Φ0〉 that has

non-zero overlap with |Ψ(R, 0)〉

limτ→∞|Ψ(R, τ)〉 = lim

τ→∞

∑i=0

e−(Ei−ET)τ |Φi〉〈Φi|Ψ(R, 0)〉

= limτ→∞

e−(E0−ET)τ |Φ0〉〈Φ0|Ψ(R, 0)〉.(2.35)


The higher terms will decay exponentially faster since En > E0 ∀n 6= 0. For

ET = E0 the function |Ψ(R, τ)〉 converges to the ground-state wave function

|Φ0(R)〉 regardless of the choice of the initial wave function |Ψ(R, 0)〉

limτ→∞|Ψ(R, τ)〉 ∝ |Φ0(R)〉. (2.36)

This fundamental property of the projector e−(H−ET)τ is the basis of the DMC

technique [43]. The DMC method follows the evolution of an initial many-body

state |Ψ(R, 0)〉 in imaginary time, until long enough time passes and only the

contribution of the ground state to the many-body wave function dominates ac-

cording to Eq. (2.35).

2.3.1 Green’s Function

To follow the evolution of the Schrodinger equation in imaginary time we will use

the Green’s function formalism.

The solution of the imaginary-time Schrodinger equation Eq. (2.30) in integral

form is given by

〈R|Ψ(τ)〉 =

∫dR′〈R|e−(H−ET)τ |R′〉〈R′|Ψ(0)〉, (2.37)

and it can be written as

Ψ(R, τ) =

∫dR′G(R|R′; τ)Ψ(R′, 0). (2.38)

Here, Ψ(R′, 0) is the wave function at the initial time τ = 0 and we have in-

troduced the Green’s function G(R|R′; τ), also known as the imaginary-time

propagator from R′ to R

G(R|R′; τ) = 〈R|e−(H−ET)τ |R′〉. (2.39)

The Green’s function is subject to the boundary condition at the initial time

τ = 0

G(R|R′; 0) = δ(R−R′). (2.40)

In general, we do not know the exact Green’s function for all times τ . How-

ever, the Green’s function is known in the limit of a short propagation time,


G(R|R′; ∆τ), where ∆τ is a small imaginary time-step

Ψ(R, τ + ∆τ) =

∫dR′G(R|R′; ∆τ)Ψ(R′, τ), (2.41)

and then Eq. (2.38) can be solved in a step by step process

Ψ(R, τ) = limM→∞

∫dR1dR2 · · · dRMG(R|RM; ∆τ)G(RM|RM−1; ∆τ)

· · ·G(R2|R1; ∆τ)Ψ(R1, 0).

(2.42)

According to Eq. (2.42) an approximation to the final state Ψ(R, τ) is obtained

by applying M times the short-time Green’s function to the intial state Ψ(R1, 0).

Before giving an explicit expression for the short-time Green’s function we

are going to introduce the importance sampling technique. In this technique, we

introduce a guiding wave function that is independent of the imaginary time.

2.3.2 Importance Sampling

Solving Eq. (2.28) is usually inefficient, mainly because of the presence of the

potential V (R), which can diverge when to particles are very close. This leads

to large variance and low convergence when calculating the expectation values of

observables. To overcome these limitations one can use the importance sampling

technique.

In the importance sampling procedure one consider the imaginary-time evo-

lution of the mixed distribution f(R, τ), which is given by the product,

f(R, τ) = ΨT(R)Ψ(R, τ), (2.43)

of the wave function Ψ(R, τ), which satisfies the Schrodinger equation Eq. (2.30),

and a trial wave function ΨT(R), which is imaginary-time independent. The

trial wave function ΨT(R) is designed from the available knowledge of the exact

ground-state wave function.

The imaginary-time evolution of f(R, τ) can be obtained by multiplying

Eq. (2.29) by ΨT(R). After rearranging terms, one obtains

−∂f(R, τ)

∂τ=−D∇2

Rf(R, τ) +D∇R · [F(R)f(R, τ)]

+ [EL(R)− ET] f(R, τ).

(2.44)


Here, F(R) denotes the drift force, also called the drift velocity

F(R) = 2∇RΨT(R)

ΨT(R), (2.45)

and EL(R) is the local energy Eq. (2.9)

EL(R) =HΨT(R)

ΨT(R). (2.46)

Eq. (2.44) describes a modified difussion process for the mixed distribution

f(R, τ). Notice that, the rate term is now proportional to [EL(R)− ET], unlike

the rate term in Eq. (2.30) which depends on the potential V (R). With a good

choice of ΨT(R), the local energy EL(R) remain finite even if V (R) diverges [49].

Also, notice that there is an additional term ∇R · [F(R)f(R, τ)] in Eq. (2.44).

This new term imposes a drift on the difussion process guided by ΨT(R).

The mixed distribution f(R, τ) becomes proportional to the ground-state

wave function in the limit of large enough time

f(R, τ) ∝ limτ→∞

ΨT(R)Φ0(R). (2.47)

2.3.3 Importance-Sampling Green’s Function and Short-

Time Approximation

The evolution described by Eq. (2.44) can be written as the sum of three different

operators acting on f(R, τ)

− ∂f(R, τ)

∂τ= (OK + OD + OB)f(R, τ) ≡ Of(R, τ), (2.48)

whereOK = −D∇2

R,

OD = D[∇R · F(R) + F(R) · ∇R],

OB = EL(R)− ET.

(2.49)

Here, OK , OD and OB are the kinetic, drift and branching operators, respec-

tively. This division will make easier to solve the Schrodinger equation for f(R, τ)

Eq. (2.44).

Analogously to Eq. (2.41), the formal solution of the evolution equation for


the mixed distribution f(R, τ)

f(R, τ + ∆τ) =

∫dR′G(R|R′; ∆τ)f(R′, τ), (2.50)

where G(R|R′; τ) is the importance sampling Green’s function. G(R|R′; τ) sat-

isfies the boundary condition

G(R|R′; 0) = δ(R−R′). (2.51)

The importance sampling Green’s function is given in terms of the operator O,

G(R|R′; ∆τ) = 〈R|e−O∆τ |R′〉. (2.52)

Now, we focus on giving an explicit expression for the short-time Green’s

function. A short-time approximation of the Green’s function to first order in ∆τ

is given by

e−O∆τ = e−(OK+OD+OB)∆τ = e−OK∆τe−OD∆τe−OB∆τ +O((∆τ)2). (2.53)

A second order descomposition is given by

e−O∆τ = e−OB∆τ2 e−OD

∆τ2 e−OK∆τe−OD

∆τ2 e−OB

∆τ2 +O((∆τ)3). (2.54)

Observe that, as ∆τ → 0 this will be a valid approximation. Introducing

Eq. (2.54) into Eq. (2.50) we obtain an integral equation of the mixed distribution

f(R, τ) in terms of the individual Green’s functions Gi, each one asociated to a

single operator Oi

f(R, τ + ∆τ) =

∫dR1dR2dR3dR4dR

′[GB(R|R1;∆τ

2)

× GD(R1|R2;∆τ

2)GK(R2|R3; ∆τ)

× GD(R3|R4;∆τ

2)GB(R4|R′;

∆τ

2)]f(R′, τ).

(2.55)

The next step is to solve three diferential equations, each corresponding to a

Green’s function Gi. The first diferential equation is associated with the kinetic


operator GK

− ∂GK(R|R′; τ)

∂τ= −D∇2

RGK(R|R′; τ). (2.56)

This is a diffusion equation which diffusion constant D. The evolution given by

GK corresponds to an isotropic Gaussian movement

GK(R|R′; τ) = (4πDτ)−3N/2 exp

[−(R−R′)2

4Dτ

]. (2.57)

The second diferential equation corresponds to the drift operator GD

− ∂GD(R|R′; τ)

∂τ= D∇R ·

[F(R)GD(R|R′; τ)

]. (2.58)

The Green’s function GD describes the movement due to the drift force and its

solution is

GD(R|R′; τ) = δ(R−R′(τ)), (2.59)

where R(τ) is defined by the following equations

dR(τ)

dτ= DF(R(τ)),

R(0) = R.

(2.60)

The last diferential equation is associated with the branching operator GB

− ∂GB(R|R′; τ)

∂τ= (EL(R)− ET)GB(R|R′; τ), (2.61)

and its solution is given by

GB(R|R′; τ) = exp [−(EL(R)− ET)τ ] δ (R−R′) . (2.62)

The Green’s function GB assigns a weight to R depending on its local energy.

Now that we have found the solutions to the equations of the Green’s functions

we can describe completely the stochastic realization of the DMC algorithm.

In the stochastic realization of the DMC algorithm, the mixed distribution

and its imaginary-time evolution are represented by a set of random walkers.

Walkers evolve through repeated applications of the propagators Gi, until one

obtains convergence to the ground state in the limit τ →∞.


2.3.4 DMC Stochastic Realization

In this section, we use the concepts exposed previously to give a basic version of

the DMC algorithm with importance sampling.

In the DMC method, the probability distribution at the initial time f(R, 0)

and its evolution in imaginary-time f(R, τ) is represented by a set of random

walkers. A walker is defined by the positions of all the particles in the system

in the configuration space of 3N dimensions R = {~r1,~r2, . . . ,~rN}. The set of

random walkers can be written as [41]

Rk = {Rk,α|α = 1, 2, . . . , Nw,k}. (2.63)

Here, k is the time step index, τk = k∆τ is the current time, and Nw,k is the

number of walkers which may change between steps.

The initial configuration for the DMC algorithm is drawn from some arbitrary

probability distribution. In most cases the initial configuration will be the output

from the VMC algorithm.

At the time-step k = 0, we start with an initial configuration Nw,0 of random

walkers R0 drawn from

fini(R) = f(R, 0) = |ΨT(R)|2, (2.64)

after passing a large enough Metropolis steps. An initial estimate of ET is ob-

tained from the mean of the local energies of the walkers

ET,0 =1

Nw,0

Nw,0∑α=1

EL(R0,α). (2.65)

The following algorithm is iterated M times.

1. Starting from the random walker Rk−1,α we obtain a temporary configura-

tion by appliyng the Green’s function GK Eq. (2.56). This is done for all

random walkers in Rk−1. This means that if we start with the configuration

Rk−1,α we obtain a temporary configuration R′k−1,α as

R′k−1,α = Rk−1,α + χ. (2.66)

Here, χ is an N-dimensional random vector sampled from a multivariate


Gaussian distribution with zero mean and variance σ2 = 2D∆τ .

2. Now we will apply a second Green’s function GD Eq. (2.58), which corre-

sponds to the action of the drift force. From the temporary configuration

R′k−1,α we obtain a new configuration Rk,α by doing the following steps

• R(1)k−1,α = R′k−1,α + F(R′k−1,α)∆τ

2

• R(2)k−1,α = R′k−1,α +

[F(R′k−1,α) + F(R

(1)k−1,α)

]∆τ4

• R(3)k−1,α = R′k−1,α + F(R

(2)k−1,α)∆τ

• Rk,α = R(3)k−1,α

The new configuration form a new set Rk.

We used a second order integration method called Runge-Kutta [53] to in-

tegrated the differential equation Eq. (2.60) in order to do the displacement

from R′k−1,α to Rk,α.

3. Calculate the branching probability for each walker in Rk:

wα = e−(EL(Rk,α)−EL(Rk−1,α)

2−ET

)∆τ. (2.67)

4. Calculate the branching factor for each walker in Rk:

nα = int(wα + η). (2.68)

Here, int denotes the integer part of a real number and η is a random

number drawn from the uniform distribution on the interval [0, 1). If nα = 0,

removed Rk,α from Rk. If nα ≥ 1, replace Rk,α with nα copies of iftsel in

Rk.

5. Update the estimators of energy and other observables of interest.

6. Repeat steps 1 to 5 until M time steps are reached.

For sufficiently long times the ground-state energy is given by

E0 = limτ→∞

∫dRf(R, τ)EL(R)∫

dRf(R, τ). (2.69)


The result of the stochastic process describe above is a set of walkers representing

the distribution f(R, τ). Therefore, the estimator for E0 after M times steps

is [36]

E0 =

∑Mk=1

∑Nw,kα=1 EL(Rk,α)∑M

k=1

∑Nw,kα=1 wk,α

. (2.70)

The value of ET is adjusted during the iterations to keep the size of the

walker population within a desired value. A simple formula for adjusting ET for

the iteration k + 1 is [36]

ET,k+1 = E0,k − Cln

(Nw,k

Nw,ave

), (2.71)

where C is a constant, and Nk,ave is a desired average number of walkers.

2.3.5 Convergence Analysis

The DMC algorithm gives exact results for the ground-state energy when simul-

taneously the time step ∆τ → 0 and the number of walkers Nw →∞. The use of

a finite time step ∆τ to approximate the Green’s function introduces a systematic

error bias in the calculation. To overcome this problem one can consider a short-

time Green’s function accurate to order (∆τ)2 according to Eq. (2.53) or a more

precise algorithm accurate to order (∆τ)3 as stated by Eq. (2.54). In the first

case, the energy has a linear dependence when the time step is sufficiently small.

Then, one can use several values of the time step to extrapolate the value of the

energy to the ∆τ → 0 limit. In the second case, the energy depends quadratically

on the time step. Here, the extrapolation procedure is not completely necessary

because for small ∆τ the energy converges fast to the exact value and the time

step can be chosen such that the systematic error is smaller than the statistical

error.

In Fig 2.1 we show an example of the ground-state energy E0 dependence on

the time step ∆τ for a dipolar gas. We compare a linear DMC method with a

second-order DMC method. For the linear DMC algorithm, in order to obtain

the exact energy, the extrapolation to zero time step is required. In Fig 2.1 we

observe that the slope of the line that joins the green dots is pronounced with

respect to the scale that we are using. In contrast, for the second-order DMC

technique, we notice that the changes in energy are statistically indistinguishable

in the range ∆τ = 0.01 − 0.1, the slope of the line that joins the blue dots is


less pronounced. This is a very useful feature of the second-order DMC method

which allows one to obtain exact values of the energy with less computational

effort. Besides the time step bias, the DMC algorithm presents a dependence on

the number of walkers Nw, which requires additional analysis.

Fig 2.2 shows an example of the ground-state energy E0 dependence on the

number of walkers Nw for a dipolar gas. We notice that the energy changes very

little for Nw & 1000. Therefore, using Nw ≈ 1000 to estimate the exact ground-

state energy is typically sufficient. This depends on the interaction potential and

mainly on the quality of the trial wave function. The improvement of ΨT makes

Nw decrease.

0.0 0.2 0.4 0.6 0.8∆τ

0.36

0.34

0.32

E0

Second− order DMC

Linear DMC

Figure 2.1: Ground-state energy E0 of a dipolar gas vs the time step ∆τ com-puted by using the linear and second-order DMC algorithms. The dashed linescorrespond to polynomial fits: E0(∆τ) = a0 + a1∆τ for linear DMC meyhodand E0(∆τ) = b0 + b1∆τ + b2(∆τ)2 for second-order DMC method. The unit ofenergy used is ~2/(mr2

0), with r0 = md2/~2 the dipolar length and d is the dipolemoment of an atom of mass m. The units of ∆τ are inverse of energy.


10-3 10-2 10-1

1/Nw

0.33

0.32

0.31

0.30

0.29E

0

Figure 2.2: Ground-state energy E0 of a dipolar gas vs the the inverse of the

number of walkers 1/Nw computed by using a second-order DMC algorithm with

time step ∆τ = 0.01. The dashed line correspond to the polynomial fit: E0(Nw) =

a0 + a1/Nw + a2/(Nw)2 + a3/(Nw)3. The unit of energy used is ~2/(mr20). The

units of ∆τ are inverse of energy.

2.4 Trial Wave Functions

An important part of the VMC and DMC methods is the choice of the trial wave

function. In the VMC technique, the expectation values of all observables are

evaluated with the trial wave function, therefore it determines completely the

accuracy of the results. In the DMC algorithm, the trial function affects the

efficiency of the estimations by increasing or decreasing the variance. The DMC

technique is based on energy projection, therefore, generally, the energy converges

faster as compared to other quantities. However, the non-diagonal properties can

be more sensitive to the quality of the trial wave function.

The trial wave function ΨT(R) should be a good approximation of the ground

state of the system. Also, for better computational efficiency ΨT(R) and its

gradient and Laplacian should have simple expressions since they are repeatedly

evaluated in the calculation.

The trial wave functions usually used in Quantum Monte Carlo methods are

of the form

ΨT(R) = F1(R)F2(R)S(R). (2.72)

2.4. Trial Wave Functions 27

The factor F1(R) is constructed as a product of one-body terms

F1(R) =N∏i=1

f1(ri). (2.73)

The one-body term f1(ri) depends only on the position of a single particle ri.

The choice of f1(ri) is based on the characteristics of the system under study. In

general, the one body functions are taken as the solution of the one-body problem

with an external potential Vext(ri).

The interparticle correlations are commonly described by a pair-product form

F2(R) known as the Bijl-Jastrow term and it is constructed as a product of

two-body terms

F2(R) =N∏j<k

f2(|rj − rk|). (2.74)

The two-body term depends on the distance between a pair of particles f2(|rj −rk|). Typically, for short distances the two-body function is constructed as the

solution of the two-body problem with an interaction potential Vint(|rj − rk|).The factor S(R) defines the symmetry or antisymmetry of ΨT(R) under the

exchange of two particles.

In Chapters 4, 5, and 6 we study a mixture of bosons of types A and B with

attractive interspecies AB interactions and equally repulsive intraspecies AA and

BB interactions. In this case, we use ΨS(R) as a trial wave function

ΨS(R) =

NA∏i<j

fAA(|ri − rj|)NB∏α<β

fBB(|rα − rβ|)

×[NA∏i=1

NB∑α=1

fAB(|ri − rα|) +

NB∏α=1

NA∑i=1

fAB(|ri − rα|)],

(2.75)

where NA and NB are the number of bosons of the species A and B, respectively.

We denote with Latin letters the bosons of the species A and with Greek letters

the bosons of the species B. In Eq. (2.75) we have removed the one-body terms

since there is no a external potential. The terms in the first row of Eq. (2.75)

are of the Bijl-Jastrow form and the term in the second row corresponds to the

factor S(R), in this case it is symmetric because our system consists of bosons.

The advantage of using ΨS(R) is that it is suitable for describing systems with

pairing. In particular, ΨS(R) describes well the the dimer-dimer problem.


Once the trial wave function has been chosen it is necessary to calculate its

gradient and Laplacian in order to implement the QMC method. The expressions

for the gradient and the Laplacian of ΨT(R) and ΨS(R) can be found in the

Appendices A and B, respectively.

2.5 Quantum Monte Carlo Estimators

The aim of this section is to show how the ground-state properties are computed

in the Monte Carlo algorithm.

2.5.1 Pair Distribution Function

The pair distribution function g(r1, r2) is proportional to the probability of finding

two particles at the positions r1 and r2, simultaneously. In coordinate represen-

tation, for a homogeneous system g(r1, r2) is given by

g(r1, r2) =N(N − 1)

n2

∫|Ψ(R)|2dr3 · · · drN∫|Ψ(R)|2dR , (2.76)

where n is the density of the system. In a homogeneous system the pair distri-

bution function g(r1, r2) depends only on the relative distance r = r1 − r2, with

this assumption g(r1, r2) becomes

g(r) =N(N − 1)

n2Ld

∫|Ψ(R)|2δ(r12 − r)dR∫

|Ψ(R)|2dR , (2.77)

where L is the size of the simulation box and d is the dimensionality of the system.

To improve the efficiency of the calculation we sum over all pair of particles

g(r) =2

nN

∫|Ψ(R)|2∑i<j δ(rij − r)dR∫

|Ψ(R)|2dR , (2.78)

where rij = ri − rj. In Monte Carlo, the pair distribution function is determined

by making a histogram of the distance between all pair of particles in the system.

2.5. Quantum Monte Carlo Estimators 29

2.5.2 One-Body Density Matrix

For a homogeneous system described by the many-body wave function Ψ(r1, · · · , rN)

the one body density matrix ρ(r1, r′1) is defined as

ρ(r1, r′1) = N

∫dr2 · · · drNΨ∗(r1, r2, · · · , rN)Ψ(r′1, r2, · · · , rN)∫

dr1 · · · drN |Ψ(r1, r2, · · · , rN)|2 . (2.79)

In the VMC calculations, we sample the trial wave function ΨT(R). Thus, a

variational estimation of the one body density matrix is given by

ρ(r1, r′1) = N

∫dr2 · · · drN Ψ∗T(R)

Ψ∗T(R′)|ΨT(R′)|2∫

dR|ΨT(R)|2 , (2.80)

where R = {r1, r2, · · · , rN} and R′ = {r′1, r2, · · · , rN}.Instead, in the DMC method we sample the mixed distribution f(R) =

ΨT(R)Ψ(R). Thus, a mixed estimation of the one body density matrix is given

by

ρ(r1, r′1) = N

∫dr2 · · · drN Ψ∗T(R)

Ψ∗T(R′)f(R′)∫

dRf(R). (2.81)

For a homogeneous Bose system, the condensate fraction N0/N is obtained

from the asymptotic behavior of the one body density matrix

lim|r−r′|→∞

ρ(r1, r′1)

n=N0

N, (2.82)

where N0 is the number of particles in the condensate.

2.5.3 Mixed Estimators and Extrapolation Technique

The expectation value of a given observable O is obtained from

〈O〉 =〈Ψ|O|Ψ〉〈Ψ|Ψ〉 , (2.83)

where Ψ is the wave function of the system. In a VMC calculation, the expecta-

tion values are evaluated with the trial wave function ΨT, therefore we obtain a

variational estimator

〈O〉var =〈ΨT|O|ΨT〉〈ΨT|ΨT〉

. (2.84)


A more precise estimator is obtained from a DMC calculation, where the expec-

tation values are sample for the mixed distribution f = ΨTΨ. After long enough

imaginary time propagation we have f ≈ ΨTΦ0, and the expectation value is

obtained from

〈O〉mix =〈ΨT|O|Φ0〉〈ΨT|Φ0〉

, (2.85)

where Φ0 is the ground-state wave function. The last equation is known as the

mixed estimator since it is calculated over two different states. The Eq. (2.85)

gives the exact expectation value for the Hamiltonian (i.e. the calculation of the

ground-state energy is exact) and for observables that commute with it. In the

case of operators that do not commute with H, the result obtained from Eq. (2.85)

will be biased by ΨT. In this case, it is possible to improve the description by

employing a first order correction in ΨT using the extrapolation method. In this

method, one assumes that the difference between the trial wave function ΨT and

the ground-state wave function Φ0 is small: δΨ = Φ0 − ΨT. The approximated

value of the exact estimator with a second-order error in δΨ can be written in

two forms〈O〉ext1 =2〈O〉mix − 〈O〉var +O(δΨ2),

〈O〉ext2 =〈O〉2mix

〈O〉var

+O(δΨ2).(2.86)

The main limitation of using the extrapolated estimators 〈O〉ext1 and 〈O〉ext2 is

that they depend on the quality of trial wave function ΨT used for importance

sampling. However, it is usefull to have two different estimators, as if they differ

among themselves, the difference will show the typical difference with the exact

result.

2.5.4 Pure Estimators

To overcome the limitations of the extrapolation method, one can use forward

walking techniques or similar methods to calculate pure estimators for local ob-

servables that do not commute with the Hamiltonian. The pure estimator of a

local observable O is given by

〈O〉pure =〈Φ0|O|Φ0〉〈Φ0|Φ0〉

. (2.87)

2.5. Quantum Monte Carlo Estimators 31

The natural outcome of the DMC method is instead a mixed estimator Eq. (2.85),

which differs from Eq. (2.87) by the presence of the trial wave function ΨT on one

of the sides. Nevertheless, the pure estimator can be related to the mixed one

by reweighting the observable with the quotient Φ0/ΨT calculated for the same

coordinates as the local observable

〈O〉pure =〈ΨT| Φ0

ΨTO|Φ0〉

〈ΨT| Φ0

ΨT|Φ0〉

=

⟨Φ0

ΨT

O

⟩mix

. (2.88)

According to Lie et. al. [54] the quotient Φ0/ΨT can be computed from the

asymptotic number of descendants of each of the walkers

W (R) = limτ→∞

n (R(τ)) . (2.89)

Usually, in the Monte Carlo algorithm the local observables are calculated

by taking block averages. Each block consists of M time steps or iterations.

Inside one of these blocks, after one iteration, when a walker is replicated, we

replicate its coordinates and its weight Eq. (2.89), and computed the observables

associated with it [55, 56]:

Ok,α = 〈O (Rk,α)〉mix,

Wk,α = n (Rk,α) ,(2.90)

where k is the time step index and α is an index over the number of walkers Nw,k

After a block is completed, the estimator of the observable is calculated as

〈O〉block =

∑Mk=0

∑Nw,kα=1 Wk,αOk,α∑M

k=0

∑Nw,kα=1 Wk,α

. (2.91)

After Nblock blocks, the pure estimator is given by

〈O〉pure =1

Nblock

Nblock∑j=1

〈O〉j. (2.92)

The pure estimator depends on the size M of a block. M has to be large enough

to reach the asymptotic regime given by Eq. (2.89).

Although the calculation of the pure estimators for the potential energy, den-

sity profile, pair distribution function, static structure factor, and other correla-


tion functions are routinely done to our best knowledge, the calculation of pure

coordinates never has been done. We found it convenient to store the coordi-

nates of the walker and replicate them during the branching process. After long

enough propagation time, the pure coordinates are stored and at the end of the

simulation, we have a large number of pure coordinates. At this point an average

of a local observable over them automatically becomes pure. In particular, we

find this trick to be very flexible and especially useful for the pure estimation of

all sorts of complicated correlation functions common for few-body analysis and

often involving calculations of hyperradius. For example, Fig. 5.3, Fig. 5.4 and

Fig. 5.5, were obtained by this method.

CHAPTER 3

ONE-DIMENSIONAL THREE-BOSON PROBLEM

WITH TWO- AND THREE-BODY INTERACTIONS

In this chapter, we study the three-boson problem with contact two- and three-

body interactions in one dimension. By using the diffusion Monte Carlo technique

we calculate the binding energy of two and three dimers formed in a Bose-Bose

or Fermi-Bose mixture with attractive interspecies and repulsive intraspecies in-

teractions. Combining these results with a three-body theory [57], we extract

the three-dimer scattering length close to the dimer-dimer zero crossing. In both

considered cases the three-dimer interaction turns out to be repulsive. Our re-

sults constitute a concrete proposal for obtaining a one-dimensional gas with a

pure three-body repulsion.

3.1 Introduction

The one-dimensional N -boson problem with the two-body contact interaction

g2δ(x) is exactly solvable. Lieb and Liniger [58] have shown that for g2 > 0

the system is in the gas phase with positive compressibility. McGuire [59] has

demonstrated that for g2 < 0 the ground state is a soliton with the chemical

potential diverging with N . In the case N =∞ the limits g2 → +0 and g2 → −0

are manifestly different: The former corresponds to an ideal gas whereas the latter

corresponds to collapse. Accordingly, the behavior of a realistic one- or quasi-

one-dimensional system close to the two-body zero crossing strongly depends

on higher-order terms not included in the Lieb-Liniger or McGuire zero-range

33

34Chapter 3. One-Dimensional Three-Boson Problem with Two- and Three-Body

Interactions

models. Sekino and Nishida [60] have considered one-dimensional bosons with

a pure zero-range three-body attraction and found that the ground state of the

system is a droplet with the binding energy exponentially increasing with N ,

which also means collapse in the thermodynamic limit. In Ref. [61], the authors

have argued that in a sufficiently dilute regime the three-body interaction is

effectively repulsive, providing a mechanical stabilization against collapse for g2 <

0. The competition between the two-body attraction and three-body repulsion

leads to a dilute liquid state similar to the one discussed by Bulgac [62] in three

dimensions.

The three-body scattering in one dimension is kinematically equivalent to a

two-dimensional two-body scattering [60, 63]. Therefore, the corresponding inter-

action shift depends logarithmically on the product of the scattering momentum

and three-body scattering length a3. An important consequence of this fact is

that, in contrast to higher dimensions, the one-dimensional three-body interac-

tion can become noticeable even if a3 is exponentially small compared to the

mean interparticle distance. Therefore, three-body effects can be studied in the

universal dilute regime essentially in any one-dimensional system that preserves

a finite residual three-body interaction close to a two-body zero crossing. Uni-

versality means that the effective-range effects are exponentially small and the

relevant interaction parameters are the two- and three-body scattering lengths a2

and a3, respectively.

In this chapter, we consider a two-component Bose-Bose mixture with at-

tractive interspecies and repulsive intraspecies interactions. In this system, the

interspecies attraction binds atoms into dimers while the dimer-dimer interaction

is tunable by changing the intraspecies repulsion [61]. Analytical predictions [57]

are complemented by diffusion Monte Carlo calculations of the hexamer energy,

permitting to determine the three-dimer scattering length close to the dimer-

dimer zero crossing. We perform this procedure for equal intraspecies coupling

constants and in the case where their ratio is infinite. In the latter limit, one of

the components is in the Tonks-Girardeau regime and the system is equivalent

to a Fermi-Bose mixture. We find that the three-dimer interaction is repulsive in

both cases.

3.2. The System 35

3.2 The System

In Ref. [57], the authors considered a one-dimensional system of three bosons of

mass m interacting via contact two- and three-body forces characterized by the

scattering lengths a2 and a3, respectively. They obtained the following analytical

expression for the ground and excited trimer energies

lna3κe

γ

a2

=2

κ2 − 1

[π

3√

3+

3κ2 − 1√4κ2 − 1

arctan

√2κ+ 1

2κ− 1

], (3.1)

where κ =√−mEa2/(2~) and γ = 0.577 is Euler’s constant. The Eq. (3.1) relates

the trimer binding energy E = E3 < 0 with a2 and a3. Considering the dimer

binding energy as |E2| = ~2/ma22 we obtain the following relation E3/E2 = 4κ2.

In the following we are going to use Eq. (3.1) to extract the three-dimer scattering

length close to the dimer-dimer zero crossing.

Systems where two- and three-body effective interactions can be controlled

independently are difficult to produce or engineer (see [63] and references therein).

We now discuss a model tunable to the regime of pure three-body repulsion.

Namely, we consider a mixture of one-dimensional pointlike bosons A and B of

unit mass characterized by the coupling constants

gAB = − 2~2

maAB

< 0, (3.2)

for the interspecies attraction and

gσσ = − 2~2

maσσ> 0, (3.3)

for the intraspecies repulsions. The interspecies attraction leads to the formation

of AB dimers of size aAB and energy

EAB = − ~2

ma2AB

. (3.4)

One can show [61] that the two-dimer interaction changes from attractive

to repulsive with increasing gσσ. In particular, the two-dimer zero crossing is


Interactions

predicted to take place for the Bose-Bose (BB) case at

gAA = gBB = 2.2|gAB|, (3.5)

and fo the Fermi-Bose (FB) case at

gAA = 0.575|gAB|, (3.6)

if gBB =∞.

Here, we consider three such dimers and characterize their three-dimer inter-

action by calculating the hexamer energy EAAABBB and by comparing it with the

tetramer energy EAABB on the attractive side of the two-dimer zero crossing where

the tetramer exists. The idea is that sufficiently close to this crossing the dimers

behave as pointlike particles weakly bound to each other. One can then extract

the three-dimer scattering length a3 from the zero-range three-boson formalism

[Eq. (3.1)], with m→ 2m

E2 = EAABB − 2EAB,

E3 = EAAABBB − 3EAB,(3.7)

and using the asymptotic expression for the dimer-dimer scattering length

E2 = − ~2

2ma22

, (3.8)

we obtain

a2 =~√

2m|E2|. (3.9)

3.3 Details of the Methods

In order to calculate E2 and E3, we resort to the diffusion Monte Carlo (DMC)

technique,, which was explained in Chapter 2. The importance sampling is used

to reduce the statistical noise and also to impose the Bethe-Peierls boundary

conditions stemming from the δ-function interactions. We construct the guiding

wave function ψT in the pair-product form

ψT =∏i<j

fAA(xAAij )

∏i<j

fBB(xBBij )

∏i,j

fAB(xABij ) , (3.10)

3.3. Details of the Methods 37

where xσσ′

ij = xσi − xσ′j is the distance between particles i and j of components σ

and σ′, respectively. The intercomponent correlations are governed by the dimer

wave function

fAB(x) = exp

(− |x|aAB

), (3.11)

and the intracomponent terms are

fσσ(x) = sinh

( |x|aAB

− |x|2add

)−(aσσaAB

− aσσ2add

). (3.12)

These functions satisfy the Bethe-Peierls boundary conditions,

∂fσσ′(x)

∂x|x=+0 = −f

σσ′(0)

aσσ′, (3.13)

which, because of the product form, also ensures the correct behavior of the

total guiding function ψT at any two-body coincidence. At the same time, the

long-distance behavior of fσσ(x) is chosen such that ψT allows dimers to be at

distances larger than their size. When the distance x between pairs {xA1 , x

B1 } and

{xA2 , x

B2 } is much larger than the dimer size aAB, Eq. (3.10) reduces to

ψT ∝ fAB(xAB11 )fAB(xAB

22 ) exp

(− |x|add

). (3.14)

For add � aAB, this wave function describes two dimers weakly-bound to each

other. It can be noted that the choice of the same spin Jastrow terms might

seem rather unusual as fσσ become exponentially large at large distances due

to divergence of sinh(x) function. This divergence is cured by multiplication of

exponentially small opposite-spin Jatsrow terms fAB(x). Thus, the use of the

sinh(x) function allows both to impose the physically correct long-range proper-

ties, Eq. (3.14) and the correct Bethe-Peierls boundary condition at short dis-

tances where the expansion sinh(x)→ x results in fσσ ∝ |x| − aσσ.

While aσσ′ are fixed by the Hamiltonian, we treat add as a free parameter in

Eq. (3.10). Close to the dimer-dimer zero crossing add ≈ a2 and this parameter

is related self-consistently to the tetramer energy while far from the crossing its

value is optimized according to the variational principle. It is useful to mention

that in case FB, where aBB = 0, the B component is in the Tonks-Girardeau limit

and can be mapped to ideal fermions by Girardeau’s mapping [64]. Replacing

|x| by x in the definition of fBB(x) makes ψT antisymmetric with respect to


Interactions

permutations of B coordinates.

3.4 Results

3.4.1 Binding Energies

In Fig. 3.1 (left) we show the tetramer energy EAABB as a function of the ratio

gAA/|gAB| for the Bose-Fermi and Bose-Bose mixtures. The thresholds for binding

are shown by arrows.

In Fig. 3.1 (right) we show the hexamer energy EAAABBB as a function of the

ratio gAA/|gAB| for the Bose-Fermi and Bose-Bose mixtures. We find that for

sufficiently strong intercomponent repulsion (larger gAA/|gAB|) the hexamer gets

unbound, first for the Bose-Fermi case and then for the Bose-Bose mixture.

0.0 0.5 1.0 1.5 2.0gAA/|gAB|

2.4

2.3

2.2

2.1

2.0

1.9

EA

AB

B/|E

AB|

Bose−BoseBose−Fermi

0.0 0.5 1.0 1.5 2.0gAA/|gAB|

4.5

4.0

3.5

3.0

EA

AA

BB

B/|E

AB|

Bose−BoseBose−Fermi

Figure 3.1: Tetramer EAABB (left) and hexamer EAAABBB (right) energies in unitsof the dimer energy |EAB| for Bose-Fermi and Bose-Bose mixtures as a function ofthe ratio gAA/|gAB|. The arrows show the positions of the thresholds for binding.

3.4.2 Threshold Determination

In Fig. 3.2 we show the numerical threshold determination for the tetramer and

hexamer for the Bose-Fermi and Bose-Bose mixtures. Our numerical results for

the tetramer threshold values are consistent with the predictions of Ref. [61]

(Eq. 3.5 and Eq. 3.6). We find that the hexamer threshold for the Bose-Fermi

mixture is located at gAA/|gAB| ≈ 0.575. In the case of the Bose-Bose mixture,

the hexamer threshold occurs at gAA/|gAB| ≈ 2.2. In both cases, the hexamer

thresholds coincide with the tetramer thresholds.

3.4. Results 39

0.0 0.5 1.0 1.5 2.0 2.5gAA/|gAB|

0.00

0.25

0.50

0.75

1.00

1.25

1.50√ 2( E AA

BB

EA

B−

2)

Bose−Bose

gAA

|gAB|= 2.199(3)

Bose−Fermi

gAA

|gAB|= 0.575(6)

0.0 0.5 1.0 1.5 2.0 2.5 3.0gAA/|gAB|

0.00

0.25

0.50

0.75

1.00

1.25

1.50

√ 2( E AAA

BB

B

EA

B−

3)

Bose−Bose

gAA

|gAB|= 2.25(2)

Bose−Fermi

gAA

|gAB|= 0.564(3)

Figure 3.2: Fitting procedure for the determination of the threshold values of theAABB tetramer and AABBB hexamer for Bose-Fermi and Bose-Bose mixtures.

The dashed curves correspond to the fit c1

(gAA

|gAB|− gcAA

|gcAB|

)+ c2

(gAA

|gAB|− gcAA

|gcAB|

)2

.

3.4.3 Three-Dimer Repulsion

In Fig. 3.3, we show E3/|E2| for cases BB (red squares) FB (blue circles) as a

function of δ = 1/ ln(√

2m|E2|a3/~) along with the prediction of Eq. (3.1) (solid

black). The quantity a3 is a fitting parameter to the DMC results; changing it

essentially shifts the data horizontally. We clearly see that in both cases the three-

dimer interaction is repulsive since E3/|E2| is above the McGuire trimer limit [59]

(dash-dotted line). For rightmost data points the hexamer is about ten times

larger than the dimer and the data align with the universal zero-range analytics.

For the other points we observe significant effective range effects related to the

finite size of the dimer. In the universal limit aAB � a2, the leading effective-

range correction to the ratio E3/|E2| is expected to be proportional to aAB/a2 ∝e1/δ [61]. Indeed, adding the term Ce1/δ to the zero-range prediction well explains

deviations of our results from the universal curve and we have checked that other

exponents do not work that well. We thus treat a3 and C as fitting parameters;

in case BB we obtain a3 = 0.01aAB and in case FB a3 = 0.03aAB. Both cases

are fit with C = −100 (dashed curve in Fig. 3.3). We emphasize that we are

dealing with the true ground state of three dimers. The lower “attractive” state

formally existing for these values of a2 and a3 in the zero-range model is an artifact

since it does not satisfy the zero-range applicability condition. The three-dimer

interaction is an effective finite-range repulsion which supports no bound states.


Interactions

- 0 . 3 0 - 0 . 2 5 - 0 . 2 0 - 0 . 1 5 - 0 . 1 0 - 0 . 0 5 0 . 0 0- 4 . 5

- 4 . 0

- 3 . 5

- 3 . 0

- 2 . 5

- 2 . 0

- 1 . 5

1 / l n [ ( 2 | E 2 | ) 1 / 2a 3 ]

z e r o r a n g e f i n i t e r a n g e B o s e B o s e F e r m i B o s e

E3 / |E

2|

1/ln[(2m|E2|)1/2 a3/~]

Figure 3.3: E3/|E2| vs 1/ ln(√

2m|E2|a3/~) for one-dimensional dimers. HereE2 and E3 are the tetramer and hexamer energies measured relative to the two-and three-dimer thresholds, respectively. The solid curve is the prediction ofEq. (3.1) and the dashed curve is a fit, which includes finite-dimer-size effectsinto account. The dash-dotted line is the McGuire result E3 = 4E2 for threepointlike bosons with no three-body interaction. The red squares are the DMCdata for case BB plotted using a3 = 0.01aAB and the blue circles stand for caseFB with a3 = 0.03aAB. The error bars are larger in the latter case because ofthe larger statistical noise induced by the nodal surface imposed by the Fermistatistics.

3.5 Summary

In conclusion, we argue that since in one dimension the three-body energy correc-

tion scales logarithmically with the three-body scattering length a3, three-body

effects are observable even for exponentially small a3, which significantly sim-

plifies the task of engineering three-body-interacting systems in one dimension.

We demonstrate that Bose-Bose or Fermi-Bose dimers, previously shown to be

tunable to the dimer-dimer zero crossing, exhibit a noticeable three-dimer repul-

sion. We can now be certain that the ground state of many such dimers slightly

below the dimer-dimer zero crossing is a liquid in which the two-body attraction

is compensated by the three-body repulsion [61, 62].

Our results have implications for quasi-one-dimensional mixtures. We men-

tion particularly the 40K-41K Fermi-Bose mixture which emerges as a suitable

candidate for exploring the liquid state of fermionic dimers. Here the intraspecies41K-41K background interaction is weakly repulsive (the triplet 41K-41K scatter-

3.5. Summary 41

ing length equals 3.2nm [65]) and the interspecies one features a wide Feshbach

resonance at 540G [66]. Let us identify A with 40K, B with 41K, and assume

the radial oscillator length l0 = 56nm, which corresponds to the confinement

frequency 2π × 80kHz. Under these conditions the effective coupling constants

equal gσσ′ ≈ 2a(3D)σσ′ /l

20 [67] and the dimer-dimer zero crossing at gAA = 0.575|gAB|

is realized for the three-dimensional scattering lengths a(3D)BB ≈ 3.2nm and a

(3D)AB ≈

−5.6nm. The dimer size is then ≈ 560nm and dimer binding energy corresponds

to ≈ 2π×800Hz placing the system in the one-dimensional regime. For the right-

most (next to rightmost) blue circle in Fig. 3.3, the tetramer is approximately

20 (10) times larger than the dimer and 800 (200) times less bound. Moving

left in this figure is realized by increasing |a3DAB| and thus getting deeper in the

region gAA < 0.575|gAB|. Note, however, that this also pushes the system out

of the one-dimensional regime and effects of transversal modes [68–70] become

important.

CHAPTER 4

FEW-BODY BOUND STATES OF

TWO-DIMENSIONAL BOSONS

In this chapter, we study clusters of the type ANBM with N ≤ M ≤ 3 in a two-

dimensional mixture of A and B bosons, with attractive AB and equally repulsive

AA and BB interactions. In order to check universal aspects of the problem, we

choose two very different models: dipolar bosons in a bilayer geometry (this

work) and particles interacting via separable Gaussian potentials (reported in

Ref. [71]). We find that all the considered clusters are bound and that their

energies are universal functions of the scattering lengths aAB and aAA = aBB, for

sufficiently large attraction-to-repulsion ratios aAB/aBB. When aAB/aBB decreases

below ≈ 10, the dimer-dimer interaction changes from attractive to repulsive and

the population-balanced AABB and AAABBB clusters break into AB dimers.

Calculating the AAABBB hexamer energy just below this threshold, we find an

effective three-dimer repulsion which may have important implications for the

many-body problem, particularly for observing liquid and supersolid states of

dipolar dimers in the bilayer geometry. The population-imbalanced ABB trimer,

ABBB tetramer, and AABBB pentamer remain bound beyond the dimer-dimer

threshold. In the dipolar model, they break up at aAB ≈ 2aBB where the atom-

dimer interaction switches to repulsion. The work presented in this chapter was

a collaboration [71]. I did the calculations for the dipolar clusters.

43

44 Chapter 4. Few-Body Bound States of Two-Dimensional Bosons

4.1 Introduction

Recent experiments on dilute quantum droplets in dipolar bosonic gases [9–12]

and in Bose-Bose mixtures [6–8] with competing interactions have exposed the im-

portant role of beyond-mean-field effects in weakly-interacting systems. A natural

strategy to boost these effects and enhance exotic behaviors is to make the in-

teractions stronger while keeping the attraction-repulsion balance for mechanical

stability. The most straightforward way of getting into this regime is to increase

the gas parameter na3s. However, this leads to enhanced three-body losses which

results in very short lifetimes (as it has been observed in experiments [6–12]).

Nevertheless, this regime is achievable in reduced geometries. It has been shown

that a one-dimensional Bose-Bose mixture with strongly-attractive interspecies

interaction becomes dimerized and, by increasing the intraspecies repulsion, the

dimer-dimer interaction can be tuned from attractive to repulsive [72]. Then, an

effective three-dimer repulsion has been found in this system and predicted to

stabilize a liquid phase of attractive dimers [57].

In two dimensions, a particularly interesting realization of such a strongly-

interacting, tunable, and long-lived Bose-Bose mixture is a system of dipolar

bosons confined to a bilayer geometry [73–75]. When the dipoles are oriented per-

pendicularly to the plane, there is a competing effect between repulsive intralayer

and partially attractive interlayer interactions, interesting from the viewpoint

of liquid formation. In addition, the quasi-long range character of the dipolar

interaction can produce the rotonization of its spectrum and a supersolid behav-

ior [76–84], formation of a crystal phase [30, 31], and a pair superfluid [32–34]

(see also lattice calculations of Ref. [85]). A peculiar feature of bilayer model is

the vanishing Born integral for the interlayer interaction [86],∫VAB(ρ)d2ρ = 0, (4.1)

which has led to controversial claims about the existence of a two-body bound

state [87] till it has finally been established that this bound state always exists,

although its energy can be exponentially small [88–92], consistently with Ref. [93].

Interestingly, a similar controversy seems to continue at the few-body level; it

has been claimed [94] that the repulsive dipolar tails will never allow for three-

or four-body bound states in this geometry.

In this chapter, we investigate few-body bound states in a two-dimensional

4.2. The Hamiltonian 45

mass-balanced mixture of A and B bosons with two types of interactions charac-

terized by the two-dimensional scattering lengths aAB and aAA = aBB. The first

case corresponds to the bilayer of dipoles discussed above and, in the second,

the interactions were modeled by non-local (separable) finite-range Gaussian po-

tentials [71]. By using the diffusion Monte Carlo (DMC) technique in the first

case, and the Stochastic Variational Method (SVM) in the second, we find that

for sufficiently weak BB repulsion compared to the AB attraction, aAB � aBB,

all clusters of the type ANBM with 1 ≤ N ≤ M ≤ 3 are bound. We then lo-

cate thresholds for their unbinding with decreasing aAB/aBB. By looking at the

AAABBB hexamer energy close to the corresponding threshold, we discover an

effective three-dimer repulsion, which can stabilize interesting many-body phases.

4.2 The Hamiltonian

The Hamiltonian of the system is

H =− ~2

2m

N∑i=1

∇2i −

~2

2m

M∑α=1

∇2α +

∑i<j

VAA(rij)

+∑α<β

VBB(rαβ) +∑i,α

VAB(riα) ,

(4.2)

where the two-dimensional vectors ri and rα denote particle positions of species

A and B containing, respectively, N and M atoms, VAB and VAA = VBB are the

interspecies and intraspecies interaction potentials, and m is the mass of each

particle. For the bilayer setup, we have

VAA(r) = VBB(r) =d2

r3, (4.3)

and

VAB(r) =d2(r2 − 2h2)

(r2 + h2)5/2, (4.4)

where d is the dipole moment and h is the distance between the layers. Dipoles are

aligned perpendicularly to the layers and there is no interlayer tunneling. The

potential VBB(r) is purely repulsive and is characterized by the h-independent

scattering length aBB = e2γr0 [95], where γ ≈ 0.577 is the Euler constant and r0 =

md2/~2 is the dipolar length. The interlayer potential VAB(r) always supports at


least one dimer state. Its energy reported in the Fig. 4.1 diverges for h→ 0 and

exponentially vanishes in the opposite limit [88–91]. The scattering length aAB,

which is a function of r0 and h, is ∼ aBB ∼ r0 for h ∼ r0, and exponentially large

for h � r0. In the following, we parametrize the system by specifying aBB and

aAB rather than h and r0.

In the more academic case of Gaussian interactions, the following potential

was used [71]

VAB(riα)ψ(riα) =

∫VAB(riα, r

′iα)ψ(r′iα)d2r′iα, (4.5)

and similarly for VAA and VBB, where

Vσσ′(r, r′) = Cσσ′Gξ(r)Gξ(r

′),

Gξ(r) = (2πξ2)−1 exp(−r2/2ξ2),(4.6)

and ξ is the characteristic range of the potential. An advantage of this non-local

potential is that the two-body problem can be solved analytically, giving

C−1σσ′ =

m

4π~2

[2 ln

2ξ

aσσ′− γ]. (4.7)

In the following, the ratio is varied aAB/aBB, with aBB = 1.4ξ fixed. Note that

100 101 102

aAB/aBB

10-5

10-4

10-3

10-2

10-1

100

|EA

B| /

[2/m

a2 BB]

Gaussian potentialDipolar potential

Figure 4.1: Dimer energy EAB in units of ~2/ma2BB for Gaussian (curve) and

dipolar (symbols) potentials as a function of the scattering length ratio aAB/aBB.


the available ratio is limited to aAB/aBB > 1.1.


In order to calculate the energies of the different few-body clusters with dipolar

interactions we use the diffusion Monte Carlo (DMC) method (see Chapter 2),

which leads to the exact ground-state energy of the system, within a statistical

error. This stochastic technique solves the Schrodinger equation in imaginary

time using a trial wave function for importance sampling. We choose it to be

ΨS(r1, . . . , rN+M) =N∏i<j

fAA(rij)M∏α<β

fBB(rαβ)

×[N∏i=1

M∑α=1

fAB(riα) +M∏α=1

N∑i=1

fAB(riα)

],

(4.8)

which takes into account a possible formation of AB dimers.

The intraspecies Jastrow factors are chosen as the zero-energy two-body scat-

tering solution,

fAA(r) = fBB(r) = K0(2√r0/r), (4.9)

with K0 the modified Bessel function. The interspecies interactions are de-

0 25 50 75 100 125 150r/r0

0

1

2

3

4

5

6

f σσ′ (r)

fAA

fAB

Figure 4.2: Intraspecies fAA(r) and interspecies fAB(r) wave functions.


scribed by the dimer wave function fAB(r) up to R0, calculated numerically.

The variational parameter R0 is chosen to be large enough that for distances

larger than R0 we neglected the dipolar potential and took the free scattering

solution fAB(r) = CK0(√−mEABr/~). We impose continuity of the logarithmic

derivative at the matching point R0, this condition yields to the following equality

f′AB(R0)

fAB(R0)= −√−mEAB

~K1(√−mEABR0/~)

K0(√−mEABR0/~)

. (4.10)

In Fig. 4.2 we show the intraspecies fAA(r) and interspecies fAB(r) wave functions.

In the Gaussian model, the stochastic variational method (SVM) was used.

Details of the SVM method can be found in Refs. [96, 97].

4.4 Results

4.4.1 Binding Energies

We first discuss the limit of very large aAB (large dimer size) when the inter-

action range and the intraspecies interactions can be neglected. In this case,

the problem can be treated in the zero-range approximation giving for the ABB

trimer EaBB=0ABB = 2.39EAB [98–100] and for the tetramers EaBB=0

ABBB = 4.1EAB and

EaBB=0AABB = 10.6EAB [98]. The other ANBM clusters (with 1 ≤ N ≤ M ≤ 3) are

also bound in absence of the intraspecies repulsion. In Ref. [101], the authors

calculated their binding energies (and they also updated the energies of smaller

clusters), which are reported in Table 4.1.

ANBM EaBB=0ANBM

/EAB

ABB 2.3896(1)

ABBB 4.1364(2)

AABB 10.690(2)

AABBB 28.282(5)

AAABBB 104.01(5)

Table 4.1: Energies of ANBM clusters in units of the dimer energy EAB in absenceof the intraspecies repulsion aBB = 0 [101]. The number between parenthesisindicates the error in the last digits of the corresponding value.

4.4. Results 49

The intraspecies repulsion shifts the cluster energies upwards as has been seen

for the ABB trimer [102, 103] and for the ABBB tetramer [103]. In Fig. 4.3, we

report the energies of these and bigger clusters for both the dipolar and Gaussian

interactions. Note that, even for the weakest BB repulsion shown in this figure

(aAB/aBB = 200), the clusters are significantly less bound compared to the case

of no BB repulsion. This happens since the small parameter that controls the

weakness of the intraspecies interaction relative to the interspecies one is actually

λ = 1/ ln(aAB/aBB)� 1. By contrast, effective-range corrections contain powers

of r0

√mE/~ or ξ

√mE/~ for dipolar or Gaussian interactions, respectively, which

are exponentially small in terms of λ. This explains why the two interaction

models lead to almost indistinguishable results for large aAB/aBB.

We find that for sufficiently strong intraspecies repulsion (smaller aAB/aBB)

the trimer and all higher clusters get unbound. In Fig. 4.3, the thresholds for

binding in the dipolar model are shown by arrows. We find that the tetramer

threshold is located at aAB/aBB ≈ 10 (h/r0 ≈ 1.1) and the trimer threshold,

100 101 102

aAB/aBB

6

5

4

3

2

1

EANBM / |E

AB|

ABBABBBAABBAABBBAAABBB

Figure 4.3: Energies of ANBM clusters in units of the dimer energy EAB forGaussian (curves) and dipolar (symbols) potentials as a function of the scatteringlength ratio aAB/aBB. The arrows show the positions of the thresholds for bindingin the dipolar case.


corresponding to the atom-dimer zero crossing, occurs in the regime where all

relevant length scales (scattering lengths, dimer sizes, interaction ranges) are

comparable to one another; aAB/aBB ≈ 2 (h/r0 ≈ 0.8) for the dipolar model.

The positions of the threshold and differences between the results of the two

models are better visible in Fig. 4.4 where we plot the cluster energies in units of

~2/ma2BB.

−1× 10−3

−5× 10−4

0.0

EA

BB−E

AB

(a) ABB

−1.5× 10−3

−1× 10−3

−5× 10−4

0.0

EA

BB

B−E

AB

(b) ABBB

−2× 10−3

−1× 10−3

0.0

EA

AB

BB−

2EA

B

(c) AABBB

−2× 10−4

−1× 10−4

0.0

EA

AB

B−

2EA

B

(d) AABB

100 101 102 103

aAB/aBB

−5× 10−4

0.0

EA

AA

BB

B−

3EA

B

(e) AAABBB

0.8 1.0 1.2 1.4 1.6h/r0

Figure 4.4: Binding energies of the few-body clusters EANBM −NEAB, in units of~2/ma2

BB, versus aAB/aBB, for Gaussian (curves) and dipolar (symbols) potentials.The arrows show the positions of the thresholds for binding in the dipolar case.The uppper axis is labeled in dipolar units.

4.4. Results 51

Our numerical calculations for larger clusters indicate that, depending on

whether they are balanced (M = N) or not, their unbinding thresholds coincide,

respectively, with the tetramer or with the trimer ones. To understand these

results note that close to these thresholds the clusters are much larger than the

dimer. Treating there the latter as an elementary boson D, the AABBB pen-

tamer and the ABBB tetramer can be thought of as weakly bound DDB or DBB

“trimers” characterized by a large aDB value and repulsive DD and BB interac-

tions (the DD interaction is repulsive since we are above the tetramer AABB

threshold). In the limit aDB →∞ the DD and BB interactions can be neglected

and the binding energies of the DDB and DBB composite trimers are asymp-

totically fractions of EABB − EAB [99]. The ABB trimer, ABBB tetramer, and

AABBB pentamer thresholds are therefore the same [see Fig. 4.4 (a,b,c)]. In the

same reasoning, close to the AABB tetramer crossing, the hexamer AAABBB is

a weakly-bound DDD state which splits into three dimers when the dimer-dimer

attraction changes to repulsion resulting in the same threshold value.

4.4.2 Threshold Determination

In this section, we numerically determine the threshold values of the few-body

clusters in the bilayer setup. To do this we need to know how the energy depends

on the interaction potential close to the threshold for unbinding. To find out this

energy dependency, let us review the principal properties of the two-body bound

state in one 1D, two 2D, and three dimensions 3D.

According to Quantum mechanics, a symmetric attractive well in 3D supports

a bound state of two particles only if the potential well depth V is larger than a

critical depth Vc [104]. Thus there is a threshold for a two-body bound state in 3D.

This is in contrast with the 1D and 2D cases where the dimer state is formed even

for infinitely small attraction between the two particles. Therefore, in 1D and

2D the threshold for the formation of the two-body bound state is absent [104].

In Table 4.2 we present a summary of the principal properties of the dimer state

in 1D, 2D, and 3D [105]. We notice that for a symmetric attractive well in 2D

the dimer state is weakly bound, with its energy depending exponentially on the

shallow potential −V , according to

EB ≈ ERe−2cER/V , with ER =

~2

mR2, (4.11)


1D 2D 3D

V � ER � ER > Vc ≈ ER

ψ(r > R) e−r/rB K0( rrB

)

{−log(r/rB), R� r � rB

e−r/rB , r � rB

e−r/rBr

rB RERV

RecER/V R ERV−Vc

EB = − ~2

mr2B− V 2

ER−ERe−2cER/V − (V−Vc)2

ER

Table 4.2: Bound-states in 1D, 2D and 3D for a potential well of size R and depthV . ψ(r > R) is the wave function outside the well, rB is the size of the boundstate, and EB its energy (ER = ~2/(mR2)) [105].

with c on the order of 1. This is the energy dependency we were looking for.

Although Eq. (4.11) is for two-body bound states we are going to use it for larger

clusters and let see if it works. Using the above result we propose to fit the DMC

binding energies with the function

EANBM −NEAB = E1 exp

[ −1

c1(aAB − acAB) + c2(aAB − acAB)2

], (4.12)

for aAB > acAB, where acAB, E1, c1, c2 are free parameters. The Eq. (4.12) can be

rewritten as

− 1

ln|(EANBM −NEAB)/E1|= c1(aAB − acAB) + c2(aAB − acAB)2, (4.13)

which is more convenient to fit the energies. In Fig. 4.5 we show the numerical

threshold determination for the dipolar clusters. On the left panel of Fig. 4.5 we

show the threshold fitting for the trimer ABB, tetramer ABBB, and pentamer

AABBB. On the right panel of Fig. 4.5 we show the tetramer AABB and hex-

amer AAABBB thresholds. The threshold values and the fitting parameters are

reported in Table 4.3. Our numerical results are consistent with our conclusions

of the previous secction, the bilayer setup have two thresholds, one for population-

4.4. Results 53

2 3 4 5 6aAB/aBB

0.0

0.2

0.4

0.6

0.8

1.0

1.2−

1/l

n|(E

ANB

M−NE

AB)/E

1|

acAB

aBB= 2.1± 0.01

acAB

aBB= 2.1± 0.05

acAB

aBB= 2.0± 0.1

ABBABBBAABBB

8 10 12 14 16 18aAB/aBB

0.0

0.2

0.4

0.6

0.8

1.0

−1/l

n|(E

ANB

M−NE

AB)/E

1|

acAB

aBB= 9.6± 0.2

acAB

aBB= 9.4± 0.2

AABBAAABBB

Figure 4.5: Fitting procedure for the determination of the threshold values of theANBM clusters with dipolar interactions. The dashed curves correspond to the fitc1(aAB

aBB− acAB

aBB) + c2(aAB

aBB− acAB

aBB)2. The fitting parameters are reported in Table 4.3.

ANBM E1 c1 c2 acAB/aBB

ABB 0.0004 0.42307 -0.0569969 2.1 ± 0.01

ABBB 0.00041 0.461707 -0.0665619 2.1 ± 0.05

AABBB 0.00046 0.437039 -0.042321 2.0 ± 0.1

AABB 0.000089 0.125298 -0.00547874 9.4 ± 0.2

AAABBB 0.00026 0.170496 -0.00803067 9.6 ± 0.2

Table 4.3: Fitting parameters for the threshold determination.

imbalanced clusters at acAB/aBB ≈ 2 and the second one for population-balanced

cluster at acAB/aBB ≈ 10.

4.4.3 Three-Dimer Repulsion

In the last part of Subsection 4.4.1, we have integrated out the internal degrees of

freedom of the dimers, replacing them by elementary point-like bosons. In fact,

the DD zero crossing that we observe for aAB ≈ 10aBB is a nonperturbative phe-

nomenon resulting from a competition between strong repulsive and attractive

interatomic forces among four individual atoms. These interactions are strong

since the corresponding scattering lengths are comparable to the typical atomic

de Broglie wave lengths ∼ 1/aAB. We emphasize that this cancellation is achieved

only for two dimers. For three dimers it is incomplete and there is a residual effec-

tive three-dimer force of range ∼ aAB (distance, where the dimers start touching

one another). In the many-body problem, this higher-order force may compete


with the dimer-dimer interaction (if it is not completely zero) or even become

dominant. In principle, one can also discuss higher-order effects of this type at

the DB zero crossing in a DB mixture, but they are expected to be subleading

since the DD and BB interactions remain finite. In the remainder of this section

we thus concentrate on the population-balanced case.

In order to characterize the effective three-dimer interaction, we follow the

method developed previously in one dimension (Chapter 3). Namely, we analyze

the behavior of the hexamer energy just below the tetramer threshold. If the

tetramer binding energy

EDD = EAABB − 2EAB, (4.14)

is much smaller than EAB, the dimer-dimer interaction can be considered point-

like and the relative DD wave function can be approximated by

φ(r) ∝ K0(κr), (4.15)

where κ =√−2mEDD/~2 is the inverse size of the tetramer. Similarly, the

AAABBB hexamer under these conditions reduces to the well-studied problem

of three point-like bosons [98, 106–111], according to which the ground-state

hexamer binding energy

EDDD = EAAABBB − 3EAB, (4.16)

should satisfy [110, 111]

EDDD/|EDD| = −16.5226874. (4.17)

We expect the ratio EDDD/|EDD| to reach the zero-range limit (4.17) as we ap-

proach the dimer-dimer zero crossing, i.e., as κaAB → 0. In Fig. 4.6, we plot

EDDD/|EDD| versus κaAB and indeed see a tendency towards the value (4.17) al-

though the effects of the finite size of the dimers and their internal degrees of

freedom, that we have neglected in the zero-range model, are obviously impor-

tant. The fact that the hexamer energy lies above the limit (4.17) points to an

effective three-dimer repulsive force. We note again that the values of the ratio

EDDD/|EDD| obtained for Gaussian and dipolar potentials are quite close to each

other for all values of aAB suggesting a certain universality of this problem and a

4.4. Results 55

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8aAB

16

14

12

10

8

6

4

2

ED

DD / |E

DD|

ρ0/aAB = 0.13

δ-pseudopotential with hard-core hyperradiusGaussian potentialDipolar potential

Figure 4.6: The hexamer-to-tetramer binding energy ratio EDDD/|EDD| as a func-tion of κaAB. The solid line is the result of the zero-range model with the hard-corehyperradius constraint at ρ0 = 0.13aAB [71].

relative unimportance of the long-range interaction tails.

In order to quantify the three-dimer interaction observed in Fig. 4.6, we com-

pared our results with a zero-range model with hard-core hyperradius constraint

ρ0, developed in Ref. [71].

The authors in Ref. [71] extended the model of three point-like dimers by

requiring that the three-dimer wave function vanishes at a hyperradius ρ0. For

three dimers, with coordinates r1, r2, and r3, the hyperradius is defined as

ρ =√x2 + y2, (4.18)

where

x =2r3 − r1 − r2√

3and y = r1 − r2, (4.19)

are the Jacobi coordinates. For this minimalistic model EDDD/|EDD| is a function

of the ratio κρ0, relating the three- and two-dimer interaction strengths. Kar-

tavtsev and Malykh [111] discussed the adiabatic hyperspherical method in detail

and applied it to the ρ0 = 0 limit, i.e., the case of no three-body interaction. The

authors in Ref. [71] accounted for finite ρ0, by set the hyperradial channel func-

tions to zero at ρ = ρ0. In this way, they obtained the ratio EDDD/|EDD| as a

function of κρ0 (for more details see Ref. [71]). We then treat ρ0 as a constant (in-


dependent of EDD) determined by fitting the DMC and SVM data in κaAB < 0.4

range. By minimizing χ2 we obtain ρ0 = 0.13aAB.

The inclusion of the three-body hard-core constraint, even corresponding to

numerically very small κρ0, leads to a spectacular deviation from Eq. (4.17).

This interesting effect is due to an enhancement of the three-dimer interaction

by strong two-dimer correlations.

Promising candidates for observing the predicted cluster states are bosonic

dipolar molecules characterized by large and tunable dipolar lengths which, at

large electric fields, tend to r0 = 5 × 10−6m for 87Rb133Cs [112, 113], r0 = 2 ×10−5m for 23Na87Rb [114, 115] and r0 = 6× 10−5m for 7Li133Cs [116]. Fermionic87Rb40K [117, 118] and 23Na40K [119–121] molecules (r0 = 7 × 10−7m and r0 =

7 × 10−6, respectively) could be turned into bosons by choosing another isotope

of K. The interlayer distance, fixed by the laser wavelength, has typical values of

h ≈ (2 − 5) × 10−7m, which is thus sufficient for observing the few-body bound

states that we predict for ratios h/r0 > 0.8.

The next step in this work is to generalize these findings to the many-body

problem when a new scale (density n) comes into play. It is important to under-

stand how the two- and three-body effects correlate with each other as one passes

through the dimer-dimer zero crossing. Although we find no qualitative difference

between the dipolar and Gaussian models in our few-body results, the long-range

tails will be important when the quantity nr0 becomes comparable to the inverse

healing length (which is where the dipolar condensate becomes rotonized). For

bilayer dipoles the relevant region of parameters is close to the dimer-dimer zero

crossing, which we predict to be at h/r0 ≈ 1.1. In Chapter 6, we studied the

many-body problem of dipolar bosons in a bilayer geometry.

4.5 Summary

To summarize, we have studied few-body clusters ANBM with N ≤ M ≤ 3

in a two-dimensional Bose-Bose mixture using different (long-range dipolar and

short-range Gaussian) intraspecies repulsion and interspecies attraction models.

In both cases, the intraspecies scattering length aAA = aBB is of the order of the

potential ranges, whereas we tune aAB by adjusting the AB attractive potential

(or the interlayer distance in the bilayer setup). We find that for aAB � aBB all

considered clusters are (weakly) bound and their energies are independent of the

4.5. Summary 57

interaction model. As the ratio aAB/aBB decreases, the increasing intraspecies

repulsion pushes the clusters upwards in energy and eventually breaks them up

into N dimers and M −N free B atoms. In the population balanced case (N =

M) this happens at aAB/aBB ≈ 10 where the dimer-dimer attraction changes to

repulsion. By studying the AAABBB hexamer near the dimer-dimer zero crossing

we find that it very much behaves like a system of three point-like particles

(dimers) characterized by an effective three-dimer repulsion. A dipolar system

in a bilayer geometry can thus exhibit the tunability and mechanical stability

necessary for observing dilute liquids and supersolid phases.

CHAPTER 5

QUANTUM HALO STATES IN TWO-DIMENSIONAL

DIPOLAR CLUSTERS

The purpose of the present chapter is to study the ground-state properties of

loosely bound dipolar clusters composed of two to six particles in a two-dimensional

bilayer geometry. We investigate whether halos, bound states with a wave func-

tion that extends deeply into the classically forbidden region [122, 123], can occur

in this system. The dipoles are confined to two layers, A and B, with dipolar

moments aligned perpendicularly to the planes. The binding energies, pair corre-

lation functions, spatial distributions, and sizes are calculated at different values

of the interlayer distance by using the diffusion Monte Carlo method. We find

that for large interlayer separations the AB dimers are halo states and follow-

ing a universal scaling law relating the energy and size of the bound state. For

ABB trimers and AABB tetramers, we find two very distinct halo structures. For

large values of the interlayer separation, such halo states are weakly bound and

the typical distances between BB and AB dipoles are similar. However, for the

deepest bound ABB and AABB clusters, and as the clusters approach the un-

binding energy threshold, we find a highly anisotropic structure, in which the AB

pair is on average several times closer than the BB pair. Similar symmetric and

asymmetric structures are observed in pentamers and hexamers, both being halo

states, thus providing halos with the largest number of particles ever observed or

predicted before.

59

60 Chapter 5. Quantum Halo States in Two-Dimensional Dipolar Clusters

5.1 Introduction

One of the most remarkable aspects of ultracold quantum gases is their ver-

satility, which permits to bring ideas from other areas of physics and imple-

ment them in a clean and highly controllable manner. Some of the examples of

fruitful interdisciplinary borrowings include Efimov, states originally introduced

in nuclear physics and observed in alkali atoms [124–126], lattices created with

counter-propagating laser beams [127, 128] as models for crystals in condensed

matter physics, Bardeen-Cooper Schrieffer (BCS) pairing theory first introduced

to explain superconductivity and later used to describe two-component Fermi

gases [129, 130]. In the present chapter, we exploit the tunability of ultracold

gases to create halo states with a number of atoms never achieved before. Orig-

inating in nuclear physics [131–133], halo dimer states have been studied and

experimentally observed in ultracold gases [134].

A halo is an intrinsically quantum object and it is defined as a bound state

with a wave function that extends deeply into the classically forbidden region [122,

123]. These states are characterized by two simultaneous features: a large spa-

tial size, due to that extension of the bound state, and a binding energy which

is much smaller than the typical energy of the interaction. One of the most

dramatic examples of a halo system, experimentally known, is the Helium dimer

(4He2), which is about ten times more extended than the size of a typical diatomic

molecule [134].

While most of the theoretical and experimental studies of halos have been

carried out in three dimensions [135–139], there is an increasing interest in halos

in two dimensions (2D) [108, 109, 122]. In fact, two dimensions are especially

interesting as halos in 2D have different properties [122] of the 3D ones. A

crucial difference between 3D and 2D geometries is that lower dimensionality

dramatically enhances the possibility of forming bound states. If the integral of

the interaction potential V (r) over all the space is finite and negative,

Vk=0 =

∫V (r)dr < 0, (5.1)

this is always sufficient to create a two-body bound state in 2D but not neces-

sarily in 3D, where the potential depth should be larger than a critical value.

Furthermore, the energy of the bound-state is exponentially small in 2D and it

5.1. Introduction 61

can be expressed as [140]

E = − ~2

(2ma2)exp

[−~2|Vk=0|

2πm

], (5.2)

where a is the typical size of the bound state. An intriguing possibility arises

when such integral is exactly equal to zero,

Vk=0 = 0, (5.3)

since a priori it is not clear if a bound state exists. This situation exactly happens

in a dipolar bilayer in which atoms or molecules with a dipolar moment d are

confined to two layers separated by a distance h and the interaction between

particles of different layer is given by

V (r) =d2(r2 − 2h2)

(r2 + h2)5/2. (5.4)

The vanishing Born integral has first lead to conclusions that the two-body bound

state disappears when the distance between the layers is large [87] although later it

was concluded that the bound state exists for any separation [88–92], consistently

with Ref. [93]. A peculiarity of this system is that the bound state is extremely

weakly bound in the limit h → ∞. That is, a potential with depth V (r = 0) =

−d2/h3 and width h would be expected to have binding energy equal to

E = −~2/(2ma2) exp(−const · r0/h), (5.5)

where r0 = md2/~2 is the characteristic distance associated with the dipolar

interaction and m is the particle mass. Instead, the correct binding energy [90, 91]

E = −4~2/mh2 exp(−8r20/h

2 +O(r0/h)), (5.6)

is much smaller as it has h−2 in the exponent and not the usual h−1. This

suggests that the bilayer configuration is very promising for the formation of two-

body halo states. Moreover, the peculiarity of the bilayer problem has resulted

in the controversial claim that the three- and four-body [94] bound states never

exist in this system, and only very recently it has been shown that actually they

are formed [71].


In this chapter, we analyze the ground-state properties of dipolar few-body

bound states within a two-dimensional bilayer setup, as candidates for halo states.

In particular, we study the ground state of up to six particles occupying the A

and B layers, with A and B denoting particles in different planes. To find the

exact system properties we rely on the diffusion Monte Carlo (DMC) method

(see Chapter 2) with pure estimators ( see Subsection 2.5.4), which has been

used previously to get an accurate description of quantum halo states in Helium

dimers [137], trimers and tetramers [138, 139]. Also, we report relevant structure

properties of the clusters, such as the spatial density distributions and the pair

distribution functions for characteristic interlayer separations.

5.2 The Hamiltonian

We consider two-dimensional systems consisting from two to six dipolar bosons of

mass m and dipole moment d confined to a bilayer setup. All the dipole moments

are oriented perpendicularly to the layers making the system always stable. The

Hamiltonian of this system is

H = − ~2

2m

NA∑i=1

∇2i −

~2

2m

NB∑α=1

∇2α +

∑i<j

d2

r3ij

+∑α<β

d2

r3αβ

+∑iα

d2(r2iα − 2h2)

(r2iα + h2)5/2

, (5.7)

where h is the distance between the layers. The terms of the Hamiltonian (5.7)

are the kinetic energy of NA dipoles in the bottom layer and NB dipoles in the

top layer; the other two terms correspond to the intralayer dipolar interactions

of NA and NB bosons; and the last accounts for the interlayer interactions. The

in-plane distance between pairs of bosons in the bottom (top) layer is denoted

by rij(αβ) = |ri(α) − rj(β)|, and riα = |ri − rα| stands for the distance between the

projections onto any of the layers of the positions of the α-th and i-th particles.

We use the characteristic dipolar length r0 = md2/~2 and energy E0 = ~2/(mr20)

as units of length and energy, respectively.

Dipoles in the same layer are repulsive, with an interaction decaying as 1/r3.

However, for dipoles in different layers the interaction is attractive for small

in-plane distance r and repulsive for larger r. In other words, a dipole in the

bottom layer induces attractive and repulsive zones for a dipole in the top layer.

Importantly, the area of the attractive cone increases with the distance between

layers h, making the formation of few-body bound states more efficient.



To investigate the structural properties of the dipolar clusters we use a second-

order DMC method (see Chapter 2) with pure estimators (see Subsection 2.5.4).

We use the same trial wave function as in Section 4.3.

5.4 Results

5.4.1 Structure of the Bound States

We first analyze the structure of few-body clusters, composed by up four particles,

as a function of the interlayer separation h. To this end, we calculate the pair

distribution function gσσ′ (r), which is proportional to the probability of finding

two particles at a relative distance r (see Subsection 2.5.1). In the case of the

ABB trimers and AABB tetramers, we also determine the ground-state density

distributions for different values of the interlayer separation.

AB Dimer

The AB dimer is strongly bound for h . r0 and its energy decays exponentially

in the opposite limit [91] of large interlayer separation. In order to understand

how the cluster size changes with h/r0, we show in Fig. 5.1 the interlayer pair

distributions gAB(r) (left) and the dipolar potential VAB(r) (right), for three values

of h/r0. The strong-correlation peak of gAB at r/r0 = 0 is due to the interlayer

attraction VAB(r) at short distances. For the cases shown in Fig. 5.1 we notice

that gAB are very wide in comparison to the dipolar potential and interlayer

distance h/r0 reflecting the exponential decay of the bound state. The tail at

large distances becomes longer as the interlayer distance increases.

ABB Trimer

The ABB trimer is bound for large enough separation between the layers h/r0 >

0.8 while, for smaller separations, it breaks into a dimer and an isolated atom

[71]. The trimer binding energy is vanishingly small for h ≈ hc, with hc ' 0.8r0,

and it becomes larger as h is increased, reaching its maximum absolute value at

h/r0 ≈ 1.05. Then, it vanishes again in the limit of h → ∞ [71]. We report

the intralayer and interlayer pair distributions, gBB(r) and gAB(r), respectively,


0 2 4 6 8 100.0

1× 10−1

2× 10−1g A

B(r

) h/r0 = 1.05 (a)

AB

0 10 20 30 400.0

1× 10−2

2× 10−2

g AB(r

) h/r0 = 1.3 (b)

0 50 100 150 200 250 300r/r0

0.0

2× 10−4

4× 10−4

g AB(r

) h/r0 = 1.6 (c)

2

1

0

VA

B(r

)

1.0

0.5

0.0

VA

B(r

)0.4

0.2

0.0

VA

B(r

)

Figure 5.1: Interlayer pair distributions gAB(r) (left) and dipolar potentialsVAB(r) (right) for AB and for three values of the interlayer distance h/r0 = 1.05,1.3 and 1.6. Notice the different scales in the x axis.

in Fig. 5.2 for strongly- (a, b) and weakly-bound (c, d) trimers. We observe that

the gAB distributions are very wide in comparison to h, similarly to what has been

observed in Fig. 5.1 for dimers. The same-layer distribution gBB vanishes when

r/r0 → 0 as a consequence of the strongly repulsive dipolar intralayer potential at

short distances. As r increases, gBB exhibits a maximum, next it monotonically

decreases with r/r0. For a weakly-bound trimer (h/r0 = 1.6), both gAB and gBB

produce long tails at large distances.

The trimer is weakly bound close to the threshold, h → hc, and for large

interlayer separation, h → ∞, but its internal structure in those two limits is

significantly different. This can be seen in Fig. 5.3 (a , b), where we plot the

trimer ground-state spatial distribution for h/r0 = 1.05 and 1.6. The heat-map

plot is shown as a function of the distance between two dipoles in the same layer

|~rB1 −~rB

2 | (horizontal axis) and the minimal 3D-distance between dipoles in differ-

5.4. Results 65

0 2 4 6 8 100.00

0.05

0.10

g AB(r

)

h/r0 = 1.05 (a)ABB

0 100 200 300 4000.00000

0.00005

0.00010

g BB(r

)

h/r0 = 1.05 (b)

0 50 100 150 200 250 3000.0000

0.0001

0.0002

0.0003

g AB(r

)

h/r0 = 1.6 (c)

0 500 1000 1500 2000 2500 3000r/r0

0

2

4

6

g BB(r

)

1e 7

h/r0 = 1.6 (d)

0 10 20 30 400.0

1× 10−2

2× 10−2

g AB(r

)

h/r0 = 1.3 (e)

AABB

0 100 200 300 4000.0

1× 10−5

2× 10−5

g BB(r

)

h/r0 = 1.3 (f)

0 50 100 150 200 250 3000.0

1× 10−4

2× 10−4

3× 10−4

g AB(r

)

h/r0 = 1.6 (g)

0 500 1000 1500 2000 2500 3000r/r0

0.0

5× 10−7

1× 10−6

g BB(r

)

h/r0 = 1.6 (h)

Figure 5.2: Interlayer and intralayer pair distributions, gAB(r) and gBB(r), forABB (a, b, c, d) and AABB (e, f, g, h) clusters, and for different values of theinterlayer distance h/r0.

ent layers min{|~rA1 − ~rB

1 |, |~rA1 − ~rB

2 |} (vertical axis). For large separation between

layers, shown in Fig. 5.3 (b) for h/r0 = 1.6, the distances between AB and BB

atoms are all of the same order, revealing an approximately symmetric structure.

However, by decreasing the distance between layers the particle distribution be-

comes significantly asymmetric. For h/r0 = 1.05 (Fig. 5.3 (a)), we observe that

the trimer spatial distribution is elongated: two dipoles in different layers are

close to each other while the third one is far away. Regardless of the interlayer

separation, the pair AB is, on average, closer than the BB pair. As the system

approaches the threshold value (h/r0 = 0.8), the trimer becomes more extended

and eventually breaks into a dimer and an single atom.


0 50 100 150 2000

25

50

75

100 ABB

h/r0 = 1.05 (a)

0 500 1000 1500 2000|~rB

1 −~rB2 |/r0

0

250

500

750

1000

min

[|~ rA−~ r

B 1|,|~ r

A−~ r

B 2|]/r 0

h/r0 = 1.6 (b)

0

500

1000

1500

2000

0 100 200 300 400 500 6000

100

200

300 AABB

h/r0 = 1.3 (c)

0 500 1000 1500 2000|~rA

1 −~rA2 |/r0

0

250

500

750

1000

min

[|~ rA 1−~ r

B 1|,|~ r

A 1−~ r

B 2|]/r 0

h/r0 = 1.6 (d)

0

500

1000

1500

2000

Figure 5.3: Heatmap plot representing the spatial structure of the ground state forABB trimer (a, b) and AABB tetramer (c ,d) for different values of the interlayerdistance. The distance between two dipoles in the same layer is plotted in thehorizontal axis and the minimum distance between dipoles in different layers isshown in the vertical axis.

AABB Tetramer

As we have shown in the previous section, an ABB trimer breaks into a dimer and

an atom when h ≈ hc. Here, we address the structure properties of the balanced

case for a tetramer, in which the number of A and B atoms is the same. The

AABB tetramer is weakly bound for large values of h/r0. When the distance

between layers decreases, the tetramer becomes unbound at h/r0 ≈ 1.1 and splits

into two AB dimers [71].

The pair distributions gAB and gBB for AABB are shown in Fig. 5.2 (e, f, g, h)

for two characteristic values of the interlayer distance h/r0. We observe a behavior

that is similar to that previously reported for ABB. That is, both gAB and gBB

are compact for the deepest bound state (h/r0 = 1.3 for AABB), and become

diffuse, showing long tails at large distances when it turns to a weakly-bound

state (h/r0 = 1.6).

The ground-state spatial distributions for the symmetric tetramer are shown

in Fig. 5.3 (c, d). We observe that for large separation h, i.e., when the tetramer

is weakly bound, it has large spatial extension and the distances between AA

and AB pairs are of the same order. As the interlayer separation is progressively

decreased, the tetramer size decreases and its structure becomes anisotropic. In

this case, the distance between dipoles in the same layer is several times larger

5.4. Results 67

than the distance between dipoles in different layers. When the tetramer ap-

proaches the threshold for unbinding (h/r0 ≈ 1.1) the cluster becomes even more

elongated and eventually, it breaks into two AB dimers.

5.4.2 Quantum Halo Characteristics

A halo is a quantum bound state in which particles have a high probability to

be found in the classically forbidden region, outside the range of the interaction

potential. The key characteristic of a halo is its extended size and tiny binding

energies. To classify a system as a halo state, one typically introduces two scaling

parameters with which the size and the energy are compared. The first parameter

is the scaling length R. For two-body systems, one commonly chooses R as

the outer classical turning point. The second parameter is the scaling energy

µBR2/~2, where µ is the reduced mass and B is the absolute value of the ground-

state energy of the cluster. The size of a cluster is usually quantified through

its mean-square radius 〈r2〉, where r is the interparticle distance. A two-body

quantum halo is then defined by the condition

〈r2〉R2

> 2, (5.8)

which means that the system has a probability to be in the classically forbidden

region larger than 50%.

The dipolar interaction in the bilayer geometry has vanishing Born integral

and thus, the AB dimer can show enhanced halo properties. In Fig. 5.4, we show

the scaling plot for the dipolar dimers, corresponding to interlayer distance from

h/r0 = 0.14 to 1.6, as indicated on the upper axis. All dimers which lie above

the halo limit 〈r2〉/R2 = 2 (horizontal line in Fig. 5.5) are halo states and follow

a universal scaling law 〈r2〉/R2 = ~2/(3µBR2), shown with a dashed line in the

figure [122]. This is exactly the case for all dimers with interlayer separations

h/r0 > 0.45. This threshold value is close to the characteristic value, h/r0 = 0.5,

for which the dimer binding energy is approximately equal to the typical energy

of the dipolar interaction EAB ≈ ~2/(mr20).

While AB dimers exist for any interlayer separation, ABB trimers and AABB

tetramers are self-bound for large h values, where AB dimers are in fact halo

states. Thus, it can be anticipated that these few-body bound states are also

halos. The sizes of three- and four-body systems are measured in terms of the


(a) (b)

10-4 10-2 100

µBR 2/ 2

10-1

101

103

105

⟨ r2⟩/R

2

AB

1.6 1.4 1.2 1.0 0.45 0.14h/r0

Figure 5.4: Bottom panel: Size vs ground-state energy scaling plot for two-body halos. The horizontal line is the quantum halo limit and the dashed one is〈r2〉/R2 = ~2/(3µBR2), which is a zero-range approximation for two-body halosin two-dimensions [122]. Top panel: Schematic representation of the AB dimerstate in two limits: (a) AB is a halo state; (b) AB is not a halo state.

mean-square hyperradius [122],

m∗ρ2 =1

M

∑i<k

mimk(ri − rk)2, (5.9)

where m∗ is an arbitrary mass unit, M is the total mass of the system, and mi is

the mass of particle i. The scaling size parameter ρ0 is given by

m∗ρ20 =

1

M

∑i<k

mimkR2ik, (5.10)

with Rik the two-body scaling length of the i− k system, which is calculated as

5.4. Results 69

(a) (b)

10-5 10-4 10-3 10-2 10-1

m∗Bρ20/

2

104

105

106

⟨ ρ2⟩/ρ

2 0

ABB

AABB

AABBB

AAABBB

Figure 5.5: Bottom panel: Size vs ground-state energy scaling plot for three-up to six-body halos. The solid line corresponds to the case of a ABB trimerwith delta interactions and without intraspecies repulsion. Top panel: Schematicrepresentacion of the ABB trimer state in two limits: (a) h→∞; (b) h→ hc.

the outer classical turning point for the i− k potential. We choose Rik equal to

zero for repulsive potentials. The condition for three- and four-body quantum

halos is now〈ρ2〉ρ2

0

> 2. (5.11)

The dependence of the scaled size on the scaled energy for ABB and AABB

are shown in Fig. 5.5. We find a non-monotonic behavior, in clear contrast with

the dependence observed in the dimer case (see Fig. 5.4). That is, the cluster

size decreases with increasing energy and reaches a minimum and then it starts

to grow again. The minima correspond to the deepest bound states [71]. This

resurgence appears as the clusters approach to the thresholds, where trimers

eventually break into a dimer and an atom, and tetramers into two dimers. We


10-6 10-5 10-4

|EANBM−NEAB|

mr20

2

103

104

105

106

107m

∗⟨ ρ2⟩

/r

2 0

2

mr20

|EANBM−NEAB|−1

ABB

AABB

AAABB

AAABBB

Figure 5.6: Cluster size vs ground-state binding energy (in dipolar units) forthree- up to six-body. The dashed line corresponds to ~2

mr20|EANBM −NEAB|−1.

want to emphasize that all the trimers and tetramers analized in Fig. 5.5 are halo

states, although they are organized in significantly different spatial structures.

On the left side of the minima, the clusters are almost radially symmetric and

all the interparticle distances are of the same order. However, at the minima and

on the right side of the minima, the cluster structures are elongated and highly

asymmetric.

The solid line in Fig. 5.5 corresponds to the case of a trimer with contact

interactions and without intraspecies repulsion. We observe that the size of the

trimer with contact interactions is much smaller than the dipolar trimer, which

indicates that repulsion has an important role in the size and energy of the clus-

ters. In the limit of exponentially small energy, large interlayer distance, the

dipolar trimer should follow the solid line.

The AABBB pentamer and AAABBB hexamer are self-bound and are man-

ifestly halo states. Their mean square size has a similar behavior to the one

observed before for the trimer and tetramer, that is a minimum corresponding to

the larger binding energy which separates a regime of nearly symmetric particle

distribution form another one, more elongated, and thus asymmetric.

In Fig. 5.6 we plot the cluster size m∗〈ρ2〉/r20 as a function of the binding

energy |EANBM −NEAB|mr2

0

~2 in dipolar units. We notice that the cluster size is a

5.5. Summary 71

0.8 1.0 1.2 1.4 1.6h/r0

105

106

m∗⟨ ρ2⟩

/r

2 0

ABB

AABB

AABBB

AAABBB

Figure 5.7: Cluster size (in dipolar units) vs h/r0 interlayer distance for three-up to six-body halos.

function of the binding energy m∗〈ρ2〉/r20 ∼ |EANBM −NEAB|−1, as indicated by

the black dashed line, and not of the total energy.

Until now, we have shown the size of the clusters as a function of the total

and binding energies. Another possibility is to plot the size of the clusters as a

function of the interlayer separation h/r0, as we can see in Fig. 5.7. Here, for small

values of h/r0 we see a clear separation between population-imbalanced (M 6= N)

and population-balanced clusters (M = N). This is a direct consequence of what

we found in the previous chapter. That is, in Chapter 4 we find that the few-body

bound states in the bilayer geometry have two unbinding thresholds, depending

on whether they are balanced or not. The first one is at h/r0 ≈ 0.8 for population-

imbalanced ABB trimer and AABBB pentamer. The second one is at h/r0 ≈ 1.1

for population-balanced AABB tetramer and AAABBB hexamer.

5.5 Summary

We used the diffusion Monte Carlo method to study the ground-state properties

of few-body dipolar bound states in a two-dimensional bilayer setup. We have

studied clusters composed by up to six particles, for different values of the in-

terlayer distance, as candidates for quantum halo states. In the case of dimers,

we find that for values of the interlayer separation larger than h/r0 = 0.45 the


clusters are halo states and they follow a universal scaling law. In the cases of

trimers up to hexamers, we find two very different halo structures. For large

values of the interlayer separation, the halo structures are almost radially sym-

metric and the distances between dipoles are all of the same scales. In contrast, in

the vicinity of the threshold for unbinding, the clusters are elongated and highly

anisotropic. Importantly, our results prove the existence of stable halo states

composed of up to six particles. To the best of our knowledge this is the first

time that halo states with such a large number of particles are predicted and

observed in a numerical experiment. Indeed, the addition of particles to a two

or three body halo states typically makes them shrink towards a more compact

liquid structure. This particular bilayer geometry is the reason of our distinct

results. We hope that these results will stimulate experimental activity in this

setup, composed by atoms with dominant dipolar interaction, to bring evidence

of these remarkably quantum halo states.

CHAPTER 6

QUANTUM LIQUID OF TWO-DIMENSIONAL

DIPOLAR BOSONS

In this chapter, we investigate the ground-state phase diagram of two-dimensional

dipoles confined to a bilayer geometry by using many-body quantum Monte Carlo

methods. The dipoles are considered to be aligned perpendicularly to the parallel

layers. We find a rich phase diagram that contains quantum phase transitions

between liquid, solid, atomic gas, and molecular gas phases. We predict the for-

mation of a novel liquid phase in which the bosons interact via purely dipolar

potential and no contact potential is required to stabilize the system. The liquid

phase, which is formed due to the balance between an effective dimer-dimer at-

traction and an effective three-dimer repulsion, is manifested by the appearance

of a minimum in the equation of state. The equilibrium density is given by the

position of the minimum of the energy and it can be controlled in a wide range

by the interlayer distance. From the equation of state, we extract the spinodal

density, below which the homogeneous system experiences a negative pressure

and breaks into droplets. Our results offer a new example of a two-dimensional

interacting dipolar liquid in a clean and highly controllable setup.

6.1 Introduction

Quantum liquids are self-bound systems, in which competing repulsive and at-

tractive interparticle interactions mechanically balance the system. The effects

of quantum mechanics and quantum statistics, such as the indistinguishability

73

74 Chapter 6. Quantum Liquid of Two-Dimensional Dipolar Bosons

of elementary particles, play an important role [141] in the description of these

systems. One of the most known quantum liquids is superfluid Helium, which

played a revolutionary role in low-temperature quantum physics. The interaction

between Helium atoms is characterized by a finite-range potential that has two

main features. At large distances, the potential has an attractive long-range van

der Waals tail that tends to hold the atoms together. Instead, at short distances,

the potential has a strongly repulsive core that prevents the liquid from collapse.

Recent experiments, motivated by a previuos theoretical proposal [5], have

enabled the experimental observation of a qualitatively different type of liquid,

quantum droplets in a mixture of Bose-Einstein condensates [6–8] and in dipolar

bosonic gases [9–12]. These quantum droplets are self-bound clusters of atoms

possessing a density that is several orders of magnitude more dilute than liquid

Helium. Both in two-component and dipolar droplets, the system collapses ac-

cording to the mean-field theory, thus the stability of the liquid state is a genuinely

quantum many-body effect.

Dipolar liquids were first experimentally observed. However, the precise de-

scription of the system is complicated because their stability is an interplay be-

tween dipolar attraction and short-range repulsion. Therefore, there is a strong

dependence on the short-range details of the interaction potential. Instead,

dipoles in a bilayer geometry may serve as a simpler and cleaner system in which

no short-range repulsion needs to be used. If the dipolar moments of the bosons

are oriented perpendicularly to the parallel layers, there is a competing effect be-

tween repulsive intralayer and partially attractive interlayer interactions, which

can produce interesting few- and many-body states, in particular liquids. For

example, a solid and a pair superfluid phases were characterized in Refs. [32, 33]

using exact Monte Carlo simulations.

Our results in Chapter 4 for few-body bound states of dipolar bosons confined

to a bilayer geometry predicts that a dimer-dimer attraction plus an effective

three-dimer repulsion can stabilize a many-body liquid state. It is therefore an

open challenge to determine the existence, formation mechanism, and properties

of the self-bound many-body system in this geometry.

In this chapter, we study a two-dimensional system of dipolar bosons confined

to the bilayer geometry. We calculate the ground-state phase diagram as a func-

tion of the density and the separation between layers by using exact quantum

Monte Carlo methods. The key result of our work is the prediction of a homo-


geneous liquid in this system. The liquid is stable in a wide range of densities

and interlayer values. We find that the critical interlayer separation at which the

liquid to gas transition happens is the same as the threshold value at which the

effective interaction between dimers changes from attractive to repulsive and a

tetramer is formed in a four-body problem. We characterize the liquid by cal-

culating its equation of state, the condensate fraction, and the equilibrium and

spinodal densities.

6.2 The Hamiltonian

We consider N bosons of mass m and dipole moment d confined to two parallel

layers separated by a distance h. It is assumed that the dipolar moment of each

boson is aligned perpendicularly to the planes by the external field. Also, we

suppose that the confinement to each plane is so tight that there is no interlayer

tunneling and no excitations of the higher levels of the tight confinement are

possible. The Hamiltonian of this system is given by

H = − ~2

2m

NA∑i=1

∇2i −

~2

2m

NB∑α=1

∇2α +

∑i<j

d2

r3ij

+∑α<β

d2

r3αβ

+∑iα

d2 (r2iα − 2h2)

(r2iα + h2)

5/2, (6.1)

where Latin (Greek) indices run over each of NA (NB) dipoles in the bottom

(top) layer. The first two terms in the Hamiltonian (6.1) correspond to the boson

kinetic energy, the next two terms are the intralayer dipolar interaction, which

is always repulsive and falls off with a power-law 1/r3. The last term is the

interlayer potential which is attractive for small values of r but repulsive for large

values of r, where r is the in-plane distance between dipoles. The interlayer

potential always supports at least one dimer state. The binding energy diverges

for h → 0 and exponentially vanishes in the opposite limit [88–91]. The dipolar

length r0 = md2/~2 is used as a unit of length.


To investigate the ground-state properties of Hamiltonian (6.1) we employ the

diffusion Monte Carlo (DMC) method, which was explained in Chapter 2. In this

work, we use two guiding trial wave functions, the first one describes two-body


correlations between individual dipoles

ΨJ(r1, . . . , rN) =∏i<j

fAA(rij)∏α<β

fBB(rαβ)∏i,α

fAB(riα), (6.2)

and the second one takes into account a possible formation of AB dimers,

ΨS(r1, . . . , rN) =

NA∏i<j

fAA(rij)

NB∏α<β

fBB(rαβ)

×[NA∏i=1

NB∑α=1

fAB(riα) +

NB∏α=1

NA∑i=1

fAB(riα)

].

(6.3)

Both choices result in a comparable DMC energy, while the convergence is differ-

ent.

The intraspecies correlations at short distances, r < Rmatch, are modeled by

the zero-energy two-body scattering solution

fAA(r) = fBB(r) = C0K0

(2√r0/r

), (6.4)

with K0(r) the modified Bessel function and Rmatch a variational parameter [142].

For distances larger than Rmatch we choose

fAA(r) = fBB(r) = C1exp

[−C2

r− C2

L− r

], (6.5)

which describes phonons [143]. The constants C0, C1 and C2 are fixed by imposing

continuity of the function and its first derivative at the matching distance Rmatch,

and also that fAA(L/2) = 1, with L the length of the squared simulation box. The

interspecies correlations fAB(r) are taken as the solution of the two-body problem

up to R1 and imposing the boundary condition f′AB(R1) = 0. For distances larger

than the variational parameter 0 < R1 < L/2 we set fAB(r) = 1. In Fig. 6.1 we

show the intraspecies fAA(r) and interspecies fAB(r) wave functions.

For simplicity, we assume a population-balanced system NA = NB = N/2,

where N is the total number of dipoles. In order to approximate the properties

of extended systems, we perform the simulations in a square box with side length

L and impose periodic boundary conditions. The total density of the system is

defined as n = N/L2.

6.4. Results 77

0 25 50 75 100 125r/r0

0

1

2

3

4

5

6

f σσ′ (r)

fAA

fAB

Figure 6.1: Intraspecies fAA(r) and interspecies fAB(r) wave functions.

6.4 Results

6.4.1 Finite Size Effects

The dipolar potential is quasi-long ranged in two dimensions, therefore its trunca-

tion at L/2 produces large finite-size corrections. It is possible to diminish them

by adding the contribution of the tail energy to the output of the DMC energy.

For a bilayer system of dipoles the potential energy tail is given by the ex-

pression

Etail =1

2L2

∫ ∞L/2

VAA(r)gAA(r)2πrdr +1

2L2

∫ ∞L/2

VBB(r)gBB(r)2πrdr

+ L2

∫ ∞L/2

VAB(r)gAB(r)2πrdr,

(6.6)

we denote by gAA = gBB and gAB the intraspecies and interspecies pair distribu-

tion functions, respectively. Substituting the interaction potential expressions

VAA(r) = VBB(r) =d2

r3and VAB(r) =

d2 (r2 − 2h2)

(r2 + h2)5/2, (6.7)


in Eq. (6.6) we obtain

Etail(n, L)

L2=

∫ ∞L/2

[d2

r3gAA(r) +

d2

r3gBB(r) + 2

d2 (r2 − 2h2)

(r2 + h2)5/2gAB(r)

]πrdr. (6.8)

An approximate value of Eq. (6.8) is obtained by substituting gAA(r) → n2A,

gBB(r)→ n2B and gAB(r)→ nAnB, which leads to

Etail = 2πd2L

[n2

A + n2B +

2nAnBL3

(4h2 + L2)3/2

]. (6.9)

The total density of the system is n = N/L2, with N = NA + NB the total

number of particles. The density of each component is one half of the total

density, nA = nB = n/2. We now substitute these relations into Eq. (6.9)

Etail

N=πd2n3/2

√N

+πd2N(

4h2 + Nn

)3/2. (6.10)

In units of ~2/mr20 we get

Etail

N

mr20

~2=π (nr2

0)3/2

2√N

+πN[

4(hr0

)2

+ Nnr2

0

]3/2. (6.11)

This significantly reduces the finite size dependence on the simulation box length

in the energy, while the residual dependence is eliminated by a fitting procedure.

After adding the tail energy Etail to the DMC energy EDMC, we extrapolate

the energy E(N) = EDMC +Etail to the thermodynamic limit value Eth using the

fitting formula

E(N) = Eth +C

N1/2, (6.12)

where C is a fitting parameter.

In Fig. 6.2, we show an example of the finite-size study for the energy. In

it, we consider a liquid phase with density nr20 = 0.001033 and h/r0 = 1.2. We

observe that the energy dependence on the number of particles scales as 1/√N ,

contrary to the law 1/N , usual for fast decaying potentials. We find that our

fitting function describes well the finite-size dependence. The same procedure is

repeated for all the densities for the gas and liquid phases.

6.4. Results 79

0.00 0.05 0.10 0.15 0.201 /

√N

−4.7× 10−4

−4.6× 10−4

−4.5× 10−4

−4.4× 10−4

−4.3× 10−4

E / N

, [2/mr

2 0]

Figure 6.2: An example of the finite-size dependence for the energy in the liquidphase at the dimensionless density nr2

0 = 0.001033 and h/r0 = 1.2. Symbols,DMC energy (with added the tail energy, Eq. (6.11)); solid line, fit Eth +C/

√N .

6.4.2 Equation of State

A possible existence of a gas or a liquid phase can be determined from the equation

of state. The dependence of the energy on density is reported in Fig. 6.3 for

different values of the interlayer distance. We have added the contribution of the

tail energy Eq. (6.11) and performed the extrapolation to the thermodynamic

limit. We observed that for h/r0 = 1.05, the energy per particle monotonically

increases as the density is increased. Thus, the smallest energy is obtained at

vanishing density and this behavior corresponds to a gas phase. However, a

drastically different behavior is observed as the interlayer separation is increased.

That is, the energy per particle becomes negative and develops a minimum at a

finite density for h/r0 ≥ 1.15. The position of the minimum corresponds to the

equilibrium density. This behavior demonstrates the presence of a homogeneous

liquid phase. The balance of forces necessary to stabilize the liquid comes from

a dimer-dimer attraction and an effective three-dimer repulsion, as proposed in

Chapter 4. Without the repulsive three-dimer force the system would behave as

an attractive gas of dimers as shown by the dashed line in Fig. 6.3. We found

that the interlayer critical value for the liquid to gas transition (h/r0 ≈ 1.1) is

the same as the threshold value for the four-body bound state of dipolar bosons,

when the tetramer breaks into two dimers (Chapter 4).


0.0 1× 10−3 2× 10−3 3× 10−3

nr20

−2× 10−4

0.0

2× 10−4

4× 10−4

(E / N

− E

2 /

2), [

2/m

r2 0]

h/r0 = 1.15

h/r0 = 1.05

(a)

0.0 5× 10−5 1× 10−4 1.5× 10−4 2× 10−4

nr20

−1.6× 10−5

−1.2× 10−5

−8× 10−6

−4× 10−6

0.0

(E / N

− E

2 /

2), [

2/m

r2 0]

h/r0 = 1.7

h/r0 = 1.6

h/r0 = 1.53

Unstable

(b)

Figure 6.3: Energy per particle E/N with half of the dimer binding energy E2/2subtracted as a function of the dimensionless density nr2

0 for different valuesof the interlayer distance h/r0. The dashed line corresponds to the mean fieldapproximation for an attractive molecular gas E/N = −π~2n/4mln[na2

dd], whereadd is the dimer-dimer scattering length. The solid lines are polynomial fits tothe energies.

6.4.3 Phase Diagram

The equations of state are used to extract the equilibrium neq and spinodal ns

densities, which are defined by the conditions of the vanishing first and second

derivatives∂E/N

∂n= 0 and

∂2E/N

∂n2= 0, (6.13)

respectively. The resulting ground-state phase diagram is reported in Fig. 6.4 as

a function of the total density nr20 and of the interlayer distance h/r0. The self-

6.4. Results 81

0.0 0.5 1.0 1.5h/r0

10-6

10-4

10-2

100

102

104

nr

2 0

(a)

Liquid

Droplets

Atomic gas

Molecular gas

Crystal

Crystal

1.0 1.2 1.4 1.6 1.8h/r0

10-5

10-4

10-3

nr

2 0

(b)

nsneq

Figure 6.4: Ground-state phase diagram. The dashed and dotted curves are theBogoliubov approximations for a 2D Bose-Bose mixture with attractive inter-species and repulsive intraspecies short-range interactions [144].

bound many-body phases are formed for large interlayer separation, h/r0 > 1.1.

Below the spinodal curve (dotted line) the homogeneous liquid is unable to sus-

tain an increasing negative pressure and becomes unstable with respect to droplet

formation. The stable liquid phase appears above the spinodal curve. Remark-

able, this phase exists in a wide range of densities and interlayer values, which are

experimentally accessible. The equilibrium density (dashed line) can be adjusted

by changing the separation between the layers: neq decreases as h/r0 increases.


For large separations h, the liquid energy is greatly decreased and in this weakly-

interacting regime it is possible to make a comparison with the predictions of

Bogoliubov theory [144] for zero-range potential (see Fig. 6.4b). The best agree-

ment is found for the smallest equilibrium and spinodal densities (i.e., for the

largest h) for which the dipolar potential is better approximated by the s-wave

scattering length. The gaseous and self-bound phases are separated by the thresh-

old h/r0 ≈ 1.1 at which the effective dimer-dimer interaction switches sign [71]

(Chapter 4) from repulsion (gas) to attraction (liquid and droplets). The gaseous

regime features a second-order phase transition between atomic and molecular

gas phases which on a qualitative level occurs when the molecule binding energy

is similar to the chemical potential. In the molecular gas phase, the atomic con-

densate is absent while the molecular one is finite [32]. In the atomic gas, the

atomic condensate is present and the system features a strong Andreev-Bahskin

drag between superfluids in different layers [34]. The gas phase shows a quan-

tum phase transition from an atomic to a molecular superfluid as the interlayer

distance decreases [32].

As the density is increased, the potential energy starts to dominate and a

triangular crystal is formed. For large separation between layers, two independent

atomic crystals are formed and the phase transition occurs when the density per

layer reaches the same critical value as in a single-layer geometry, nAr20 = nBr

20 ≈

290 [142, 145]. In the limit of small interlayer separations, a single molecular

crystal is formed at the density nr20 ≈ 9.

6.4.4 Depletion of the Condensate

In order to have a more complete description of the liquid and gas phases we have

calculated the condensate fraction using the one-body density matrix (OBDM),

which is defined as [146]

n(1)(r, r′) = 〈Ψ†(r)Ψ(r′)〉, (6.14)

where Ψ†(r)(Ψ(r)) is the field operator that creates (annihilates) a particle at the

point r. For a homogeneous system, the condensate fraction is obtained from the

asymptotic behavior of the OBDM

n(1) (|r− r′|)|r−r′|→∞ →N0

N. (6.15)

6.4. Results 83

This behaviour is often referred to as off-diagonal long-range order, since it in-

volves the nondiagonal components (r 6= r′) of the OBDM. In Fig 6.5 (a) we

show an example of the OBDM for different numbers of particles. We notice an

important dependence on the total number of particles, that is, the N0/N value

decreases as the total number of particles increases. From the VMC and DMC

results of the OBDM, we obtained the extrapolated values of N0/N , using the

extrapolation technique (Section 2.5.3). In Fig 6.5 (b) we plot the extrapolated

0 50 100 150 200r/r0

0.75

0.80

0.85

0.90

0.95

n(1

) (r)

(a)

N=30N=60N=90N=120

0.00 0.01 0.02 0.03 0.041 / N

0.0

0.2

0.4

0.6

0.8

1.0

N0 / N

(b)

Figure 6.5: (a) Example of the one-body density matrix for h/r0 = 1.2 at theequilibrium density and for different number of particles. (b) An example of thefinite-size dependence of N0/N .


values of N0/N as a function of 1/N . In order to remove the finite-size effects,

we extrapolate the condensate fraction value N0/N to the thermodynamic limit

using the fitting formula a0 + a1/N , where a0 and a1 are fitting parameters.

0.00 0.05 0.10 0.15 0.20 0.251/|ln(na2

0 )|

0.0

0.2

0.4

0.6

0.8

1.0

N0 / N

(a)

1− 1/|ln(na20 )|

Liquid

0.0 0.5 1.0 1.5 2.0h/r0

0.0

0.2

0.4

0.6

0.8

1.0

N0 / N

(b)1− 1/|ln(na20 )|

LiquidGas

Figure 6.6: Quantum depletion of the condensate. The blue circles correspondto the liquid phase at the equilibrium density. The solid line corresponds to thequantum depletion of short-range potentials having s-wave scattering length a0

in two dimensions 1−1/|ln(na20)|. (a) Condensate fraction N0/N as a function of

1/|ln(na20)|. (b) Condensate fraction N0/N as a function of the interlayer distance

h/r0, the green triangles correspond to the gas phase at the dimensionless densitynr2

0 = 0.001.

6.4. Results 85

In Fig. 6.6 (a) we show the condensate fractionN0/N as a function of 1/|ln(na20)|,

for the dipolar liquid at the equilibrium density. In the dilute limit, we find a good

agreement with the quantum depletion 1/| ln(na20)| calculated in a Bogoliubov

theory for short-range potentials. The depletion of the condensate is present in

interacting bosonic systems in which, due to the interactions, a portion of bosons

are in a non-zero momentum state even at zero temperature. The equilibrium

density has a strong dependence on the interlayer separation h (see Fig. 6.4).

For liquids formed at separations h & 1.6 the perturbative result is expected to

hold. In the Fig. 6.6 (b) we report the condensate fraction as a function of the

interlayer separation h. The liquid exists for large separation between the layers

h. As h is decreased the equilibrium density grows up until it reaches nr20 ≈ 10−3

at h/r0 ≈ 1.1 where a phase transition to a gas happens. For smaller separations,

the liquid does not exist and we show the condensate fraction in the gas with

the density fixed to nr20 = 10−3. The condensate fraction rapidly drops to zero

signaling a phase transition from atomic to molecular gas.

6.4.5 Polarization

The different phases present in the system can be further characterized by calcu-

lated the ground-state energy dependence on small polarization. This dependence

can be linear or quadratic depending on the molecular or atomic nature of the

system, respectively. This can be obtained by slightly unbalanced the number

of particles in the bottom NA and top NB layers, while keeping fixed the total

number of particles NA +NB. The polarization is defined as

P =NA −NB

NA +NB

, with |P | � 1. (6.16)

For an atomic condensate the ground-state energy dependence on small po-

larization is quadratic

E(P ) = E(0) +N(n/2χs)P2, (6.17)

where E(0) is the ground-state energy of the balanced system and χs is the spin

susceptibility associated with the dispersion of spin waves of the magnetization

density nt − nd with speed of sound cs =√n/mχs. In this case the low-lying

excitations are coupled phonon modes of the two layers [32].


0.00 0.05 0.10 0.15 0.20P

0.0

0.05

0.1

0.15(E

(P) −

E(0

)) / E

(P)

h/r0 = 1.2

Liquid

h/r0 = 0.8

Molecular Gas

h/r0 = 1.0

Atomic gas

Figure 6.7: Ground-state energy E(P ) as a function of the polarization P for threevalues of h/r0 in the molecular gas (nr2

0 = 0.001), atomic gas (nr20 = 0.001), and

liquid phase (equilibrium density).

For a molecular superfluid phase the ground-state energy is a linear function

of the polarization

E(P ) = E(0) +N∆gapP, (6.18)

in this case an energy ∆gap is needed to break a pair and spin excitations are

gapped [32].

Examples of the different behaviors of E(P ) are reported in Fig. 6.7 for three

values of h/r0 corresponding to the molecular gas, atomic gas, and liquid phases.

We notice that E(P ) is a quadratic function of P for the liquid state, therefore

the liquid is a liquid of atoms and not a liquid of molecules or dimers.

6.5 Summary

We have shown that a dipolar bilayer possesses a rich phase diagram with quan-

tum phase transitions between gas and solid phases (known before), and a liquid

phase (newly predicted). Remarkably, the liquid state, which results from the bal-

ance of a dimer-dimer attraction and an effective three-dimer repulsion, exists in

a wide range of densities and interlayer separations which are experimentally ac-

cessible. From the equations of state, we extracted the spinodal and equilibrium

densities, which are controllable through the interlayer distance. The equilib-

6.5. Summary 87

rium density decreases as the interlayer distances increases, allowing access to

ultra-dilute liquids in a stable setup.

CHAPTER 7

PHASES OF DIPOLAR BOSONS CONFINED TO A

MULTILAYER GEOMETRY

7.1 Introduction

In a classical crystal at low temperature all the atoms are strongly localized

around their equilibrium lattice positions. In contrast, in a quantum crystal,

the atoms move around the equilibrium lattice positions and exchanges between

few particles occur with frequency [44]. A quantum crystal is then defined as a

crystal in which the zero point motion of an atom about its equilibrium position

is a large fraction of the near neighbor distance [147]. This large displacement

is a consequence of the light-weight particles and the weakness of the long-range

forces between the atoms [44].

The most known quantum solids are Hydrogen and Helium. The solidification

of 4He in 1926 initiated the experimental study of quantum solids. 4He solidifies

at the temperature limit T → 0K under a pressure of P ' 25 bar [147].

An interesting property that quantum solids can exhibit is supersolidity, a

state of matter predicted in 1969 for Andreev and Lifshitz in which crystalline

order and Bose-Einstein condensation coexists [44]. A supersolid flows without

friction but its particles form a crystalline lattice.

Ultracold dipolar gases provide a powerful platform to study highly non-trivial

quantum phenomena, in particular to study solid and supersolid states of matter.

Recent experimental and theoretical studies have shown that dipolar condensates

within a pancake- [77, 148–154] and cigar-like [155] geometries can undergo a

89

90 Chapter 7. Phases of Dipolar Bosons Confined to a Multilayer Geometry

phase transition from a Bose-Einstein condensate (BEC) to a state that has

supersolid properties.

An interesting confined system in two dimensions is a layer of dipolar bosons.

If the dipolar moments of the bosons are oriented perpendicularly to the layer, the

dipole-dipole interaction is repulsive. When this happens, the system undergoes a

quantum phase transition from a gas to a solid phase as the density increases [30,

31].

Adding a second parallel layer makes the system richer. In a bilayer setup,

there is a competing effect between repulsive intralayer and partially attractive

interlayer interactions, which can produce exotic few- and many-body states. For

example, a gas and a pair superfluid phases were characterized in Refs. [32, 33]

using exact quantum Monte Carlo simulations. Furthermore, when the interlayer

distance approaches to zero the system forms a molecular crystal and for large

values of the interlayer distance an independent atomic crystal is formed in each

layer [145].

In this chapter, we study a system of dipolar bosons confined to a multilayer

geometry by using exact many-body quantum Monte Carlo methods. The mul-

tilayer geometry consists of equally spaced two-dimensional layers. We consider

the case in which the dipoles are aligned perpendicularly to the parallel layers.

This system is predicted to have a rich collection of many-body phases due to the

anisotropic and quasi long-range dipole-dipole interaction between the bosons.

We calculated the ground-state phase diagram as a function of the density, the

separation between layers, and the number of layers. The key result of our work

is the existence of three phases: atomic gas, solid, and gas of chains, in a wide

range of the system parameters. We find that the density of the solid phase

decreases several orders of magnitude as the number of layers in the system in-

creases. Furthermore, we calculated the pair distribution functions for the three

phases.

7.2 The Hamiltonian

We study an N -particle system of dipolar bosons of mass m and dipole moment

d confined to an M -layer geometry. The three-dimensional confining structure

is formed of M two-dimensional parallel layers separated by a distance h. The

dipolar moment of each boson is considered to be perpendicular to the layers and


there is no interlayer tunneling. In Fig. 7.1 we show a schematic representation

of the multilayer geometry. The Hamiltonian of this system is given by

H =N∑i=1

− ~2

2m∇2i +

N∑i<j

Vij(|ri − rj|, l), (7.1)

where ri is the position vector of particle i. The first term in the Hamiltonian (7.1)

corresponds to the dipole kinetic energy. The second term is the dipolar interac-

tion between particle i and j

Vij(rij, l) = d2r2ij − 2l2h2

(r2ij + l2h2)5/2

, l = 0, 1, 2, . . . ,M − 1. (7.2)

Here, lh denotes the interlayer separation and rij = |ri−rj| stands for the distance

between the projections onto any of the layers of the positions of the i-th and

j-th particles. For l 6= 0 the dipolar interaction called interlayer interaction, is

attractive for small values of r but repulsive for large values of r, where r is

the in-plane distance between dipoles. For dipoles in the same layer the dipolar

interaction is always repulsive

Vij(ri,j, 0) =d2

r3ij

, (7.3)

where we set l = 0. The dipolar length r0 = md2/~2 is used as a unit of length.

In Fig. 7.2 we show the dipolar potential as a function of r/r0 for different values

of the interlayer distance lh, with h/r0 = 1.0. Notice that the attractive part of

Figure 7.1: Schematic representation of dipoles confined to a multilayer geometry.


0 1 2 3 4r/r0

2.0

1.5

1.0

0.5

0.0

0.5

1.0V(r,l

)

l= 0

l= 1

l= 2

l= 3

Figure 7.2: Dipolar potential V (r, l) as a function of r/r0 for different values ofthe interlayer separation lh, with h/r0 = 1.0.

the interlayer potential for l = 2 and 3 are much weaker than for l = 1.

For the Hamiltonian Eq. (7.1) we anticipate to find three phases: atomic gas,

solid, and gas of chains. This comes from considering previous studies in a dipolar

layer [30, 31] and bilayer [32, 33, 145].


To investigate the ground-state properties of Hamiltonian Eq. (7.1) we employ the

diffusion Monte Carlo (DMC) method, which was described in Chapter 2. Here,

we use different trial wave functions appropriate for the description of various

phases present in the system. In total we describe three phases with different

symmetry: gas, solid, and gas of chains.

7.3.1 Gas Trial Wave Function

To describe the gas phase we chose a trial wave function of the Bijl-Jastrow form

Eq. (2.74)

ΨGas(r1, . . . , rN) =N∏j<k

f2(|rj − rk|). (7.4)


The wave function ΨGas has translational symmetry. This symmetry is present in

the gas phase, while the crystal phase has broken translational symmetry due to

a regular spacing of atoms and, the gas of chains phase has additional interlayer

correlations. The two-body terms f2(|rj−rk|) in Eq. (7.4) depend on the distance

between a pair of particles. Here, the two-body terms f2(r) are taken as the

solution of the two-body problem at short distances. This solution depends on

whether the dipoles are in the same layer or not. For dipoles in the same layer

we chose the short distance part of the two-body correlations term as

f2(r) = C0K0

(2√R0r0/r

), (7.5)

up to Rmatch. Here, K0(r) the modified Bessel function, Rmatch and R0 are varia-

tional parameters [142].

For distances larger than Rmatch we chose

f2(r) = C1exp

[−C2

r− C2

L− r

], (7.6)

which take into account the contributions from other particles and describe long-

range phonons in the form established by Reatto and Chester [143]. The constants

C0, C1 and C2 are fixed by imposing continuity of the function and its first

derivative at the matching distance Rmatch, and also that f2(L/2) = 1.

For dipoles in different layers, the interlayer correlations are taken as the

solution of the two-body problem f2(r) up to R1. We also impose the boundary

condition f′2(R1) = 0. For distances larger than the variational parameter 0 <

R1 < L/2 we set f2(r) = 1.

7.3.2 Solid Trial Wave Function

The trial wave function we use to describe the quantum solid phase is of the

Nosanow-Jastrow form [44]

ΨSolid(r1, . . . , rN ; {RcI}) =

N∏j<k

f2(|rj − rk|)N∏i=1

f1(|ri −Rci |), (7.7)

where ri are the positions vectors of the particles, {RcI} are the position vectors

defining the equilibrium crystal lattice, and f1(r) and f2(r) are the one-body

and two-body correlation factors, respectively. For a two-dimensional system the


equilibrium crystal lattice is a triangular lattice. The wave function ΨSolid (7.7)

has a broken translational symmetry due to particle localization close to the

lattice sites. In two dimensiones the crystal has triangular symmetry.

The two-body correlation functions f2(r) used in ΨSolid (7.7) are of the same

form as those used to describe the gas phase ΨGas (7.4), although the specific

values of the variational parameters might differ.

The one-body terms f1(r) used in ΨSolid (7.7) are modeled by a Gaussian

function

f1(ri) = e−α|ri−Rci |2 , (7.8)

where α is the localization strength and Rci is the position of the lattice site.

Notice that, Rci are fixed by the triangular lattice. Meanwhile, α is a variational

parameter, which is chosen by minimizing the variational energy.

The trial wave function ΨSolid (7.7) is not symmetric under the exchange of

particles. Therefore, it does not give an appropriate description of off-diagonal

properties that directly depend on the Bose-Einstein statistics (for example, one

body density matrix and momentum distribution). However, ΨSolid leads to an

accurate description of the energy and diagonal properties of quantum solids

(for example, density profile, pair-distribution function, static structure factor).

Examples of symmetric trial wave functions for modeling quantum solids can be

found in Ref. [44]. In this work, we are interested in calculating the ground-state

energy and the pair distributions of the system. Thus we will use ΨSolid (7.7).

We leave for future work the use of symmetric trial wave functions to describe

the properties of our system.

7.3.3 Gas of Chains Trial Wave Function

To describe the gas of chains phase we propose the following trial wave function

ΨChains(r1, . . . , rN) =N∏j<k

f2(|rj − rk|)N∏i=1

fCM(|ri −Ri|). (7.9)

Differently from the Nosanow-Jastrow wave function, the last product in Eq. (7.9)

is not a one-body but rather is a many-body term. That is, the movement of a

single particle in that product changes M terms, with M the number of layers,

while in Nosanow form Eq. (7.7) such a movement affects only a single term (i.e.

is a one-body correlation). Here, the two-body term f2(|rj − rk|) depends on the

7.4. Details of the Gas of Chains Wave Function 95

distance between a pair of particles. The many-body term fCM(|ri−Ri|) depends

on the positions of a single particle ri and on the center of mass of the chains Ri.

Details of the construction of the trial wave function ΨChains and its derivatives

can be found in Section 7.4.

The chain center of mass Ri is given by

Ri =1

M

∑k∈Ci

rk. (7.10)

Here, k ∈ Ci means that the sum is over all k-th particles that belong to the

same chain as the i-th particle. The main difference with the solid case is that in

ΨChains (7.9) the Ri values change during the simulation while the Rci values in

ΨSolid (7.7) are fixed by the triangular lattice.

The two-body correlation functions f2(r) used in ΨChains (7.9) are the same

as those for the gas and solid phases.

The terms fCM(r) are described by a Gaussian function

fCM(ri) = e−α|ri−Ri|2 , (7.11)

where α is the localization strength.

All the variational parameters that appear in the trial wave functions ΨGas,

ΨSolid and ΨChains, are chosen by minimizing the variational energy and can have

a different value for each trial function.

7.4 Details of the Gas of Chains Wave Function

In this section, we are going to discuss a trial wave function which we employ for

the description of a gas of chains phase. Also, we will give explicit expressions

for the gradient and Laplacian of the trial function.

While the trial Bijl-Jastrow (gas) and Nosanow-Jastrow (crystal) forms are

standard and were extensively studied in the literature, within our best knowledge

this is the first time the gas of chains is calculated within the DMC method.

For methodological reasons we provide a very detailed description of how such

functions are constructed and give explicit expressions of how the derivatives are

calculated.

It is essential that the trial wave function has the same symmetry as the one


present in the phase. In the gas of chains phase, dipoles belonging to different

layers form composite bosons (chains) with no crystal ordering between them.

Each composite bosons is composed from M dipoles each belonging to a different

layer. Also, this gas fulfills the following properties:

• The chains are considered to be a composite objects and no exchange of

dipoles between them is allowed.

• The chains are flexible, each dipole in the chain can move freely in the

corresponding layer.

• The chains will not become tangled because of the repulsive intralayer in-

teraction between dipoles.

7.4.1 Construction of Trial Wave Function

A trial wave function to describe the gas of chains phase can be constructed as a

product of many-body and two-body terms as given in Eq. (7.9). The two-body

term f2(|rj−rk|) depends on the distance between a pair of particles. The many-

body term fCM(|ri −Ri|) depends on the positions of a single particle ri and on

the the chain center of mass Ri. The chain center of mass Ri itself depends on

the positions of M particles.

The chain center of mass Ri is given by

Ri ≡1

M

∑k∈Ci

rk. (7.12)

Here, M is the number of layers, and the index k runs over all k-th particles that

belong to the same chain as the i-th particle.

In order to implement the QMC algorithm we need to be able to calculate

the gradient ∇riΨChains and Laplacian ∆riΨChains of the trial wave function. The

expressions for the gradient and Laplacian of the two-body terms∏N

j<k f2(|rj−rk|)can be found in the Appendix A. Here, we are going to obtain the expressions

for the many-body terms∏N

i=1 fCM(|ri − Ri|). The gradient of the product of

many-body terms fCM(|ri −Ri|) with respect to the coordinate ri is given by

~FCM,ri=~∇ri

∏Nj=1 fCM(|rj −Rj|)∏N

j=1 fCM(|rj −Rj|)=∑k∈Ci

~∇rifCM(|rk −Rk|)fCM(|rk −Rk|)

. (7.13)

7.4. Details of the Gas of Chains Wave Function 97

Here, ~FCM,riis called the center of mass drift force and corresponds to the loga-

rithmic derivative with respect to ri.

The Laplacian ∆ri

∏Nj=1 fCM(|rj−Rj|) applied with respect to the coordinate

ri reads as

∆ri

∏Nj=1 fCM(|rj −Rj|)∏N

j=1 fCM(|rj −Rj|)=∑k∈Ci

∆rifCM(|rk −Rk|)fCM(|rk −Rk|)

−(~∇rifCM(|rk −Rk|)fCM(|rk −Rk|)

)2

+

(∑k∈Ci

~∇rifCM(|rk −Rk|)fCM(|rk −Rk|)

)2

.

(7.14)

Now we can write an expression for the center of mass contribution to the kinetic

energy

T locCM =

~2

2m

[N∑i=1

E locCM,ri −

N∑i=1

|~FCM,ri|2], (7.15)

where we have used Eq. (7.13) and we have defined the center of mass local energy

as

E locCM,ri = −

∑k∈Ci

∆rifCM(|rk −Rk|)fCM(|rk −Rk|)

−(~∇rifCM(|rk −Rk|)fCM(|rk −Rk|)

)2 . (7.16)

7.4.2 Many-Body Term: Gaussian Function

The purpose of the many-body term fCM(|ri − Ri|) in the trial wave function

ΨChains (7.9) is to describe the localization of the particles belonging to one chain

around its center of mass. The simplest form of the localization term is a Gaussian

function Eq. (7.11)

fCM(|rj −Rj|) = e−α|rj−Rj |2 , (7.17)

where α is the localization strength, rj is the position vector of the particle j and

Rj is the chain center of mass to which particle rj belongs. The chain center of

mass Rj is given by Eq. (7.12).

To calculate the center of mass contributions to the drift force ~FCM,ri(7.13)

and local energy E locCM,ri (7.16) we need expressions for the gradient and Laplacian

of the function (7.17). Let us start by calculating the gradient ~∇rie−α|rk−Rk|2


with respect to the particle ri

~∇rie−α|rk−Rk|2

e−α|rk−Rk|2=

2αM

(rk −Rk) if rk 6= ri,

2α(

1M− 1)

(ri −Ri) if rk = ri,(7.18)

where we assumed that rk ∈ Ci.

The Laplacian ∆rie−α|rk−Rk|2 with respect to the particle ri reads as

∆rie−α|rk−Rk|2

e−α|rk−Rk|2=

4α2

M2 |rk −Rk|2 − 2αM2 if rk 6= ri,

4α2(1− 1

M

)2 |ri −Ri|2 − 2α(1− 1

M

)2if rk = ri,

(7.19)

where we assumed that rk ∈ Ci.

Using Eq. (7.13) and Eq. (7.18) we obtain an expression for the center of mass

contribution to the drift force ~FCM,ri

~FCM,ri= −2α(ri −Ri). (7.20)

From here, the square of the drift force |~FCM,ri|2 reads as

|~FCM,ri|2 = 4α|ri −Ri|2. (7.21)

Using Eq. (7.16) and Eq. (7.19) we obtain that the center of mass contribution

to the local energy E locCM,ri reads as

E locCM,ri = 2α

(1− 1

M

). (7.22)

7.5 Results

In order to approximate the properties of the systems in the thermodynamic limit

we perform calculations in finite-size systems with periodic boundary conditions.

The total density of the system is defined as n = N/(LxLy), where N is the total

number of dipoles and Lx×Ly is the size of the simulation box. In the gas phase,

the simulations are performed in a square box Lx = Ly, while in the solid and gas

of chains phases the simulations are carried out in a rectangular box Lx 6= Ly,

commensurate with the crystal lattice. We ensure that each of the box sides is a

multiple of elementary cell size of a triangular lattice.

7.5. Results 99

7.5.1 Crystallization and Threshold Densities

Crystallization Density

In Ref. [31], the authors studied the ground-state phase diagram of a two-dimensional

Bose system with dipole-dipole interactions using the QMC methods. The dipoles

were constrained to move in a single plane and were polarized in the perpendic-

ular direction to the plane. The authors found a quantum phase transition from

a gas to a solid phase as the density increases. This transition was estimated to

occur at the critical density

nr20 ≈ 290, (7.23)

with r0 = r0 = md2/~2 and n = n for a single layer of dipoles.

Our system consists of dipolar bosons confined in an M -layer geometry sepa-

rated by a distance h. Now, we are going to show how Eq. (7.23) is rewritten for

the system with M layers in the limit of rigid chains. To do this, consider a chain

with M dipoles, one in each layer. When h→ 0 the chain becomes a super-dipole

with mass Mm, dipolar moment Md, and its dipolar length is given by

r0 =(Mm)(Md)2

~2= M3md

2

~2= M3r0. (7.24)

Now consider a M -layer system with N dipoles and N/M chains evenly dis-

tributed. When h → 0 the M -layer system effectively becomes to a single-layer

one, each chain becomes a super-dipole, and the number of particles changes from

N dipoles to N/M super-dipoles. As a consequence of the latter the density now

is given by

n =n

M. (7.25)

Using Eq. (7.24) and Eq. (7.25) we obtain

nr20 =

( nM

) (M3r0

)2= M5nr2

0. (7.26)

From the last relation and with Eq. (7.23) we obtain

nr20 =

290

M5. (7.27)

Notably, the solidification density has a strong dependence (fifth power) on the

number of layers. In the limit when h→ 0 an M -layer system of dipolar bosons


0.0 0.5 1.0 1.5 2.0h / r0

10-3

10-2

10-1

100

101

102

103nr

2 0M= 1

2

3

4

5

6789M= 10

limh→0

ncryr20 = 290

M 5

ntr20 = M

2(M− 1)2(h/r0)2

Figure 7.3: Schematic representation of the phase diagram of dipolar bosonsconfined to an M -layer geometry. The phase diagram is shown as a functionof the total density nr2

0, the separation between layers h/r0, and for differentnumber of layers M . The crystallization densities ncryr

20, Eq. (7.28), are shown

by the thick lines. The threshold densities ntr20, Eq. (7.33), at which classical

interaction between the top and bottom layers changes the sign are shown withdashed lines.

will crystallize at the critical density 290/M5

limh→0

ncryr20 =

290

M5. (7.28)

Such a strong dependence on M makes the multilayer geometry a very promising

setup for future observation of solidification. In Fig. 7.3 we show a schematic

representation of the phase diagram of dipolar bosons confined to an M -layer

geometry. The phase diagram is shown as a function of the total density nr20,

the separation between layers h/r0, and for different number of layers M . The

crystallization densities ncryr20 Eq. (7.28) for different values of the number of

layers are shown by the thick lines. Notice that the critical density drops several

orders of magnitude when going from M = 2 to 10 layers. This is an important

result because it tells us that a very dilute quantum solid can be obtained just

by increasing the number of layers in the system.

7.5. Results 101

Threshold Density

The dipolar interlayer potential Eq. (7.2)

Vint(r, l) = d2 (r2 − 2l2h2)

(r2 + l2h2)5/2, l = 1, 2, . . . ,M − 1, (7.29)

is anisotropic. It is attractive for small values of r but it is repulsive for large

values of r, where r is the in-plane distance between dipoles. Consider an M -layer

system of dipolar bosons in a solid phase. Here, the system forms a triangular

lattice. Now, we ask ourselves what is the threshold distance rt between a dipole

on the first layer and a dipole on the last layer and in a next neighbour lattice

site such that the dipolar interlayer potential is zero. This threshold distance rt

can be obtained from Eq. (7.29)

d2 r2t − 2(M − 1)2h2

(r2t + (M − 1)2h2)5/2

= 0, (7.30)

with l = M − 1. Now we solve for rt

r2t = 2(M − 1)2h2. (7.31)

In Fig. 7.4 we show the interlayer potential Vint(r,M − 1) as a function of r/r0

for different values of the number of layers M , with h/r0 = 1.0. The threshold

distances rt are the points where the potential crosses the horizontal axis. The

values of rt are indicated by arrows in Fig. 7.4.

Now, the mean-inter-particle distance in one layer is proportional to

〈r/r0〉 ∼1√

nr20/M

→ nr20

M∼ 1

〈r/r0〉2. (7.32)

From the Eq. (7.31) and Eq. (7.32) we obtain the threshold density nt, which

satifies Eq. (7.30)

ntr20 =

M

2(M − 1)2h2. (7.33)

The threshold density Eq. (7.33) is an approximation. Densities larger than the

threshold density ntr20 (attractive interlayer potential) will favor the formation of

a gas phase and lower densities than the threshold density (repulsive interlayer

potential) will favor the formation of the solid phase.


In Fig. 7.3 we plot the threshold density, at which the interaction potential

between the most top and most bottom layers changes its sign at the mean

interparticle distance, as a function of the interlayer separation and for different

values of the number of layers. Each of the shown lines divides the phase diagram

into two regions. The first region is above the curve, where the formation of a gas

phase is favored. The second region is below the curve where the formation of

the solid phase is facilitated. According to the prediction Eq. (7.33), it is possible

to have a solid phase at very low densities and for a wide range of h/r0.

Finally, we want to emphasize that the equations Eq. (7.28) and Eq. (7.33),

which correspond to the crystallization ncryr20 and threshold ntr

20 densities, are

approximations under some assumptions. These predictions give us a picture of

how the boundaries of the phase diagram are delimited. To calculate the exact

phase diagram it is necessary to do DMC calculations, we will do this in the next

section.

7.5.2 Dipoles within a Four-Layer Geometry

In this section, we present and discuss our DMC results for the pair distribution

functions and the ground-state phase diagram of dipolar bosons confined to a

0 1 2 3 4 5 6r/r0

0.25

0.20

0.15

0.10

0.05

0.00

0.05

Vin

t(r,M−

1)

M= 2

M= 3

M= 4

Figure 7.4: Interlayer potential Vint(r,M − 1) as a function of r/r0 for differentvalues of the number of layers M , with h/r0 = 1.0. The arrows show the positionsof the threshold distances rt for M =2, 3 and 4.

7.5. Results 103

0.0 0.5 1.0 1.5 2.0h / r0

10-3

10-2

10-1

100

101

102

nr

2 0

M= 4

Solid Gas

Chainsncryr

20

ntr20

Figure 7.5: Ground-state phase diagram of dipolar bosons confined to a four-layer system at zero temperature. The phase diagram is shown as a function ofthe total density nr2

0 and the separation between layers h/r0. The green circlescorrespond to the gas phase, the blue squares correspond to the solid phase, andthe red triangles correspond to the gas of chain phase. The crystallization ncryr

20

and threshold ntr20 densities are shown by the thick and dashed lines, respectively.

four-layer geometry.

Four-Layer Phase Diagram

The phase diagram of the dipolar multilayer is constructed is the following way.

We explore the parameter space (total density nr20 and interlayer distance h/r0)

and we calculate the ground-state energy with each of the three trial wave func-

tions: ΨGas, ΨSolid and ΨChains (see Section 7.3). The phase at each point corre-

sponds to the phase that yields the lowest energy.

The ground-state phase diagram of dipolar bosons within a four-layer geom-

etry is plotted in Fig. 7.5. The phase diagram is shown as a function of the total

density nr20 and the separation between layers h/r0. The crystallization Eq. (7.28)

and threshold Eq. (7.33) densities for M = 4 are shown by the thick and dashed

lines, respectively. We found three phases in this system: solid, gas, and gas

of chains. The gas phase and its boundaries were precisely determined, as well

as, the transitions solid-gas and chain-gas. However, a precise estimation of the

solid-chain transition location is numerally complicated, because the energies of

these phases are very similar over a wide range of the parameter space. We expect


the gas of chains phase to appear below the crystallization density ncryr20 ≈ 0.283.

The key result of the phase diagram of Fig. 7.5 is the existence of three phases:

gas, solid, and gas of chains, in a wide range of densities and interlayer distances.

This characteristic makes the experimental observation of these three phases more

feasible.

Pair Distributions

To quantify how the dipoles are spatially distributed in the gas, solid, and gas of

chains phases, we calculate the pair distribution function g(r, l) for different values

of the system parameters. The pair distribution g(r, l) is proportional to the

probability of finding two particles at the relative distance r (see Subsection 2.5.1).

Here, lh denotes the interlayer distance between particles. For example, g(r, l =

0) corresponds to the pair distribution of particles in the same layer, g(r, l = 1)

corresponds to the pair distribution of particles at a one-layer distance.

In the left panel of Fig. 7.6 we show the pair distribution functions g(r, l) for

the gas phase and for l = 0, 1, 2, 3. We have set the total density at nr20 = 2.0 and

interlayer distance h/r0 = 0.5. The same-layer distribution g(r, l = 0) vanishes

when r/r0 → 0 as a consequence of the strongly repulsive dipolar intralayer

0 1 2 3 4r/r0

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

g(r,l)

Gas g(r, l= 0)g(r, l= 1)g(r, l= 2)g(r, l= 3)

0 2 4 6 8x/r0

0

1

2

3

4

5

6

7

8

y/r 0

Figure 7.6: Left panel: A typical example of the pair distribution functions g(r, l)for the gas phase within a four-layer geometry. The gas parameters are: totaldensity nr2

0 = 2.0 and interlayer distance h/r0 = 0.5. Here, lh denotes the inter-layer distance between particles. Right panel: snapshot of the DMC simulationsfor the gas phase.

7.5. Results 105

0.0 0.5 1.0 1.5 2.0 2.5r/r0

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00g(r,l)

Solid g(r, l= 0)g(r, l= 1)g(r, l= 2)g(r, l= 3)

0 1 2 3 4 5x/r0

0

1

2

3

4

5

y/r 0

Figure 7.7: Left panel: A typical example of the pair distribution functions g(r, l)for a solid phase within a four-layer geometry. The solid parameters are: totaldensity nr2

0 = 4.0 and interlayer distance h/r0 = 0.3. Here, lh denotes the inter-layer distance between particles. Right panel: snapshot of the DMC simulationsfor the solid phase.

potential. As r increases, g(r, l = 0) exhibits a shallow maximum, next it tends

to 1, the asymptotic value of uncorrelated particles. The strong-correlation peak

of g(r, l 6= 0) at r/r0 = 0 is due to the interlayer attraction. As r increases,

g(r, l 6= 0) exhibits a minimum, next it tends to 1. In the left panel of Fig. 7.6 we

show a snapshot of the particle coordinates during the DMC simulation. Here,

the dipoles are uniformly distributed.

In the left panel of Fig. 7.7 we plot the pair distribution functions g(r, l) for

the solid phase and for l = 0, 1, 2, 3, with nr20 = 4.0 and h/r0 = 0.3. When

r/r0 → 0 we observe a behavior that is similar to that previously reported for

the gas phase. The same-layer distribution start from 0 and the different-layer

distributions have a strong correlation peak. As r/r0 increases the solid pair

distributions show a more complex structure than in the case of the gas phase.

Here, all g(r, l) show some local maxima and minima rather than tend to 1.

These extreme values are related to the parameters of the system, nr20 and h/r0.

The maxima that appear around r/r0 = 1 correspond to the mean interparticle

distance and are related to the density of the system. That is, these maxima

apper when r/r0 ≈ 1/√nr2

0/M = 1/√

4/4 = 1. In the left panel of Fig. 7.7 we


the dipoles are distributed in a triangular lattice.


0 10 20 30 40 50r/r0

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

g(r,l)

Chains g(r, l= 0)g(r, l= 1)g(r, l= 2)g(r, l= 3)

0 20 40 60 80 100x/r0

0

20

40

60

80

100

y/r 0

Figure 7.8: Left panel: A typical example of the pair distribution functions g(r, l)for a gas of chains phase within a four-layer geometry. The chains parametersare: total density nr2

0 = 0.01 and interlayer distance h/r0 = 0.9. Here, lh denotesthe interlayer distance between particles. Right panel: snapshot of the DMCsimulations for the chains phase.

In the left panel of Fig. 7.7 we plot the pair distribution functions g(r, l) for

the gas of chains phase and for l = 0, 1, 2, 3, with nr20 = 0.01 and h/r0 = 0.9.

When r/r0 → 0 we observe a similar behavior as for the gas and solid phases.

As r/r0 increases g(r, l) shows a maximum, next it tends to 1. The maxima now

appear atr/r0 ≈ 1/√nr2

0/M = 1/√

0.01/4=20. In the left panel of Fig. 7.8 we


the dipolar chains are homogeneously distributed.

7.5.3 Dipoles within an M-Layer Geometry

In this section, we present and discuss our DMC results for the phase diagram of

dipolar bosons confined to an M parallel layers at zero-temperature.

The ground-state phase diagram of dipolar bosons within an M -layer geome-

try is plotted in Fig. 7.9. The phase diagram is shown as a function of the total

density nr20, the separation between layers h/r0, and for different number of layers

M . The crystallization ncryr20 and threshold ntr

20 densities are shown by the thick

and dashed lines, respectively. The estimated critical interlayer distance for the

transition solid-gas is reported for M = 2 up to 10. For each value of M , the

critical interlayer distance is reported for one density value. Notice that, as we

7.6. Summary 107

0.0 0.5 1.0 1.5 2.0h / r0

10-3

10-2

10-1

100

101

102

103

nr

2 0

M= 1

2

3

4

5

6789M= 10

ncryr20

ntr20

Solid

Gas

Figure 7.9: Ground-state phase diagram of dipolar bosons confined to a M -layersystem at zero temperature. The phase diagram is shown as a function of thetotal density nr2

0, the separation between layers h/r0, and for different numberof layers M . The squares indicated the interlayer critical value for the transitionsolid-gas for fixed values of density and number of layers M . The crystallizationncryr

20 Eq. (7.28) and threshold ntr

20 Eq. (7.33) densities are shown by the thick

and dashed lines, respectively.

increase the number of layers the size of the phase diagram increases for the solid

phase. Also that the simple approximation (dashed lines) discussed previously

fails dramatically when M grows.

7.6 Summary

We used the diffusion Monte Carlo method to study the ground-state phase dia-

gram of dipolar bosons in a geometry formed by equally spaced two-dimensional

layers. This system is predicted to have a rich collection of phases due to the

anisotropic and quasi long-range dipole-dipole interaction between the bosons.

We predicted several quantum phase transitions occurring at zero temperature

as the density and separation between layers are changed. First, we have consid-

ered the case where there are four layers and the same number of dipoles in each

layer. We observe a number of distinct phases, including atomic gas, solid, and

gas of chains. These phases are present in a wide range of densities and interlayer

distances. The solid phase is predicted to be formed for large densities and small


interlayer distances, while the chain phase is presented for lower densities and for

a wide range of interlayer distances. The atomic gas is observed in a wide range

of densities and interlayers distances. The transitions of solid-gas and chain-gas

were determined. Whereas, the exact location of the solid-chain transition could

not be determined. Furthermore, we have considered the case where the dipoles

are confined up to ten layers. We find that the range of densities where the solid

is observed decreases several orders of magnitude with increasing the number of

layers in the system. Our results show that the dipolar multilayer system offers

a highly controllable setup for observing ultra-dilute quantum solids.

A subject of future work is the implementation of the symmetric trial wave

function to describe the solid phase. Also, the calculation of the static structure

factor and the superfluid fraction for the solid, gas, and gas of chains. The

calculation of these properties will allow us to accurately characterize the solid-

chain transition.

CHAPTER 8

CONCLUSIONS

In this Thesis, we reported a detailed study of the ground-state properties of a

set of quantum few- and many-body systems. In particular, we have studied one-

dimensional Bose-Bose and Fermi-Bose mixtures with attractive interspecies and

repulsive intraspecies contact interactions. Here, we characterized a three-dimer

repulsion. Also, we have studied a bosonic dipolar quantum system confined

to a bilayer and multilayer geometries. These setups consist of equally spaced

two-dimensional layers, which can be experimentally realized by imposing tight

confinement in one direction. We have studied the bilayer configuration with

few and many dipoles and the multilayer system with many dipoles only. In all

cases, we used Quantum Monte Carlo simulations to obtain the zero-temperature

properties.

Noticeable, we have found in the bilayer that some properties are inherited

from the few-dipole system to the many-dipole system. In the case of the few-

dipole system, we established the existence of dipolar bound states with interest-

ing properties. For example, we found that the bound states are quantum halo

states. In the case of the many-dipole system, we demonstrated the existence of

a dilute liquid phase. For the multilayer system we found an extremely dilute

solid phase. Our results show that a dipolar system in a bilayer and a multilayer

geometries offer clean and highly controllable setups for observing interesting

phases of quantum matter, such as, halo states, and dilutes liquids and solids.

Below we present the main conclusions of this Thesis.

109

110 Chapter 8. Conclusions

One-Dimensional Three-Boson Problem with Two- and Three-Body

Interactions

In Chapter 3, by using the diffusion Monte Carlo technique we calculated the

binding energy of two and three dimers formed in a one-dimensional Bose-Bose or

Fermi-Bose mixture with attractive interspecies and repulsive intraspecies inter-

actions. Combining these results with a three-body theory [57], we extracted the

three-dimer scattering length close to the dimer-dimer zero crossing. We argued

that since in one dimension the three-body energy correction scales logarithmi-

cally with the three-body scattering length a3, three-body effects are observable

even for exponentially small a3, which significantly simplifies the task of engi-

neering three-body-interacting systems in one dimension. We demonstrated that

Bose-Bose or Fermi-Bose dimers, previously shown to be tunable to the dimer-

dimer zero crossing, exhibit a noticeable three-dimer repulsion. We can now be

certain that the ground state of many such dimers slightly below the dimer-dimer

zero crossing is a liquid in which the two-body attraction is compensated by the

three-body repulsion [61, 62].

Few-Body Bound States of Two-Dimensional Bosons

In Chapter 4, we have studied few-body clusters of the type ANBM with N ≤M ≤ 3 in a two-dimensional Bose-Bose mixture of A and B bosons, with attractive

AB and equally repulsive AA and BB interactions. We considered two very

different models: dipolar bosons in a bilayer geometry and particles interacting via

separable Gaussian potentials. In both cases, the intraspecies scattering length

aAA = aBB is of the order of the potential ranges, whereas we tune aAB by

adjusting the AB attractive potential (or the interlayer distance in the bilayer

setup). We find that for aAB � aBB all considered clusters are (weakly) bound

and their energies are independent of the interaction model. As the ratio aAB/aBB

decreases, the increasing intraspecies repulsion pushes the clusters upwards in

energy and eventually breaks them up into N dimers and M −N free B atoms.

In the population balanced case (N = M) this happens at aAB/aBB ≈ 10 where

the dimer-dimer attraction changes to repulsion. The population-imbalanced

ABB trimer, ABBB tetramer, and AABBB pentamer remain bound beyond the

dimer-dimer threshold. In the dipolar model, they break up at aAB ≈ 2aBB where

the atom-dimer interaction switches to repulsion. By studying the AAABBB

hexamer near the dimer-dimer zero crossing we find that it very much behaves

111

like a system of three point-like particles (dimers) characterized by an effective

three-dimer repulsion. A dipolar system in a bilayer geometry can thus exhibit

the tunability and mechanical stability necessary for observing dilute liquids and

supersolid phases.

Quantum Halo States in Two-Dimensional Dipolar Clusters

In Chapter 5, we have studied the ground-state properties of loosely bound few-

body bound states in a two-dimensional bilayer geometry. We have investigated

whether halos, bound states with a wave function that extends deeply into the

classically forbidden region, can occur in this system. The dipoles are confined to

two layers, A and B, with dipolar moments aligned perpendicularly to the layers.

We have studied clusters composed by up to six particles, for different values

of the interlayer distance, as candidates for quantum halo states. In the case of

dimers, we find that for values of the interlayer separation larger than h/r0 = 0.45

the clusters are halo states and they follow a universal scaling law. In the cases of

trimers up to hexamers, we find two very different halo structures. For large values

of the interlayer separation, the halo structures are almost radially symmetric and

the distances between dipoles are all of the same scales. In contrast, in the vicinity

of the threshold for unbinding, the clusters are elongated and highly anisotropic.

Importantly, our results prove the existence of stable halo states composed of up

to six particles. To the best of our knowledge this is the first time that halo

states with such a large number of particles are predicted and observed in a

numerical experiment. Indeed, the addition of particles to a two or three body

halo states typically makes them shrink towards a more compact liquid structure.

This particular bilayer geometry is the reason of our distinct results.

Quantum Liquid of Two-Dimensional Dipolar Bosons

In Chapter 6, we have shown that a dipolar bilayer possesses a rich phase diagram

with quantum phase transitions between gas and solid phases (known before), and

a liquid phase (newly predicted). Remarkably, the liquid state, which results from

the balance of a dimer-dimer attraction and an effective three-dimer repulsion,

exists in a wide range of densities and interlayer separations which are experi-

mentally accessible. From the equations of state, we extracted the spinodal and

equilibrium densities, which are controllable through the interlayer distance. The

112 Chapter 8. Conclusions

equilibrium density decreases as the interlayer distances increases, allowing access

to ultra-dilute liquids in a stable setup.

Phases of Dipolar Bosons Confined to a Multilayer Geometry

In Chapter 7, we have studied the ground-state phase diagram of dipolar bosons

in a multilayer geometry formed by equally spaced two-dimensional layers. We

predicted several quantum phase transitions occurring at zero temperature as

the density and separation between layers are changed. We have considered the

case where there are four layers and the same number of dipoles in each layer.

When the dipole moment of the bosons is aligned perpendicular to the layers,

we observe three distinct phases, namely atomic gas, solid, and gas of chains.

These phases are present in wide range of densities and interlayer distances. The

solid phase is observed for large densities and small interlayer distances. While

the chain phase is presented for lower densities and for a wide range of interlayer

distances. The atomic gas is observed in a wide range of densities and interlayers

distances. The transitions of solid-gas and chains-gas were precisely determined.

However, the transition solid-chains could not be fully determined. Furthermore,

we have considered the case where the dipoles are confined to three up to ten

layers. We find that the range of densities where the solid is observed decreases

several orders of magnitude with increasing the number of layers in the system.

Our results show that the dipolar multilayer system offers a highly controllable

setup for observing ultra-dilute quantum solids.

APPENDIX A

TRIAL WAVE FUNCTION

One of the most used many-body trial wave functions ΨT(R) in QMC methods

for importance sampling is of the form

ΨT(R) =N∏i=1

f1(ri)N∏k<j

f2(|rk − rj|)S(R). (A.1)

The one-body term f1(ri) depends only on the position of a single particle ri.

The product of two-body f2(|rk − rj|) terms is known as the Bijl-Jastrow factor.

The factor S(R) defines the symmetry or antisymmetry of the trial wave function

ΨT(R) under the exchange of two particles.

In order to implement the QMC algorithm we need to calculate the gradient

∇riΨT(R) and Laplacian ∆riΨT(R) of the trial wave function. Let us start by

obtaining expressions for the gradients involved in Eq. (A.1). The gradient of the

product of one-body terms f1(rj) with respect to the coordinate ri is given by

~∇ri

∏Nj=1 f1(rj)∏N

j=1 f1(rj)=~∇rif1(ri)

f1(ri), (A.2)

and the gradient of the product of two-body terms f2(|rk − rj|) is given by

~∇ri

∏Nk<j f2(|rk − rj|)∏N

k<j f2(|rk − rj|)=∑j

~∇rif2(|ri − rj|)f2(|ri − rj|)

. (A.3)

Using the expressions shown in the Eq. (A.2) and Eq. (A.3) we obtain an ex-

113

114 Appendix A. Trial Wave Function

pression for the gradient of the trial wave function ~∇riΨT(R) with respect to the

coordinate ri

~Fri =~∇riΨT(R)

ΨT(R)=~∇riS(R)

S(R)+~∇rif1(ri)

f1(ri)+∑j


. (A.4)

Here, ~Fri is called the drift force. Now we are going to calculate the Laplacian

expressions. The Laplacian of the many-body trial wave function ∆riΨ(R) with

respect to the coordinate ri is given by

∆riΨ(R)

Ψ(R)=

∆riSS +

∆ri

∏Nj=1 f1(rj)∏N

j=1 f1(rj)+

∆ri


k<j f2(|rk − rj|)

+ 2~∇riSS ·

(∇rif1(ri)

f1(ri)+∑j

∇rif2(|ri − rj|)f2(|ri − rj|)

)

+ 2

(∇rif1(ri)

f1(ri)

)·(∑

j

∇rif2(|ri − rj|)f2(|ri − rj|)

),

(A.5)

where we have used the expresiones Eq. (A.2) and Eq. (A.3). The inner products

that appear in Eq. (A.5) are difficult to implement in the QMC code. In order

to remove these terms we considered the square of the drift force, from Eq. (A.4)

we obtain

| ~Fri |2 =

(~∇riSS

)2

+

(~∇rif1(ri)

f1(ri)

)2

+

(N∑j=1


)2

+ 2~∇riSS ·

(~∇rif1(ri)

f1(ri)+∑j


)

+ 2

(~∇rif1(ri)

f1(ri)

)·(∑

j


).

(A.6)

115

Now, we solve for the inner product terms

| ~Fri|2 −(~∇riSS

)2

−(~∇rif1(ri)

f1(ri)

)2

−(∑

j


)2

=

2~∇riSS ·

(~∇rif1(ri)

f1(ri)+∑j


)

+ 2

(~∇rif1(ri)

f1(ri)

)·(∑

j


).

(A.7)

After substituting Eq. (A.7) into Eq. (A.5) we get an expression for the Laplacian

of the trial wave function ∆riΨ(R) with respect to the coordinate ri and without

inner product terms

∆riΨ(R)

Ψ(R)=

∆riSS +

∆ri

∏Nj=1 f1(rj)∏N

j=1 f1(rj)+

∆ri


k<j f2(|rk − rj|)+ | ~Fri |2

−(~∇riSS

)2

−(~∇rif1(ri)

f1(ri)

)2

−(∑

j


)2

.

(A.8)

Now we need to calculate the expressions of the Laplacians ∆ri

∏Nj=1 f1(rj) and

∆ri

∏Nk<j f2(|rk − rj|) and then substitute them in the last equation. The Lapla-

cian ∆ri

∏Nj=1 f1(rj) is given by

∆ri

∏Nj=1 f1(rj)∏N

j=1 f1(rj)=

∆rif1(ri)

f1(ri), (A.9)

and the Laplacian ∆ri

∏Nk<j f2(|rk − rj|) reads as

∆ri


k<j f2(|rk − rj|)=

N∑j=1

∆rif2(|ri − rj|)f2(|ri − rj|)

−(~∇rif2(|ri − rj|)f2(|ri − rj|)

)2

+

(∑j


)2

.

(A.10)

Substituting the Eq. (A.9) and Eq. (A.10) into Eq. (A.8) we obtain an expres-

sion for the Laplacian of the trial wave function ∆riΨ(R) with respect to the

116 Appendix A. Trial Wave Function

coordinate ri

∆riΨ(R)

Ψ(R)=| ~Fri |2 +

∆riSS −

(~∇riSS

)2

+∆rif1(ri)

f1(ri)−(~∇rif1(ri)

f1(ri)

)2

+∑j


−(~∇rif2(|ri − rj|)f2(|ri − rj|)

)2 . (A.11)

The Eq. (A.4) and Eq. (A.11) are the equations that are implemented in the

QMC code. Now we can write an expression for the kinetic energy

T loc =~2

2m

[N∑i=1

E locS,i +

N∑i=1

E loc1,i + 2

N∑i<j

E loc2,i −

N∑i=1

|~F1,ri |2], (A.12)

where we have used Eq. (A.6) and defined

E locS,i = −∆riS

S +

(~∇riSS

)2

, (A.13)

E loc1,i = −∆rif1(ri)

f1(ri)+

(~∇rif1(ri)

f1(ri)

)2

, (A.14)

E loc2,i = −

∑j


−(~∇rif2(|ri − rj|)f2(|ri − rj|)

)2 . (A.15)

APPENDIX B

SYMMETRIC TRIAL WAVE FUNCTION

An important part of the VMC and DMC methods is the choice of the trial wave

function, which is used for importance sampling. Here, we consider a symmetric

many-body trial wave function ΨS(R), which is given by

ΨS(R) =

NA∏i<j

fAA(|ri − rj|)NB∏α<β

fBB(|rα − rβ|)

×[NA∏i=1

NB∑α=1

fAB(|ri − rα|) +

NB∏α=1

NA∑i=1

fAB(|ri − rα|)].

(B.1)

We use ΨS(R) to study a mixture of A and B bosons with attractive interspecies

AB interactions and equally repulsive intraspecies AA and BB interactions. In

Eq. (B.1), NA and NB are the number of bosons of the species A and B, respec-

tively. We denote with Latin letters the bosons of the species A and with Greek

letters the bosons of the species B.

In order to implement the QMC algorithm we need to calculate the gradient

∇riΨS(R) and Laplacian ∆riΨS(R) of the trial wave function. In the following we

are going to obtain the expressions of ∇riΨS(R) and ∆riΨS(R). To simplifying

117

118 Appendix B. Symmetric Trial Wave Function

the expressions we defined the following quantities

A ≡NA∏i<j

fAA(|ri − rj|), P1 ≡NA∏i=1

NB∑α=1

fAB(|ri − rα|),

B ≡NB∏α<β

fBB(|rα − rβ|), P2 ≡NB∏α=1

NA∑i=1

fAB(|ri − rα|).(B.2)

With the above definitions the trial wave function ΨS(R) Eq. (B.1) reduces to

ΨS(R) = AB[P1 + P2]. (B.3)

Let us now obtain expressions for the gradients involved in Eq. (B.1). The gradi-

ent of the trial wave function ~∇riΨS(R) with respect to the coordinate ri is given

by

~Fri =~∇riΨS(R)

ΨS(R)=~∇riAA

+~∇riP1 + ~∇riP2

P1 + P2

. (B.4)

Here, ~Fri is called the drift force. The term ~∇riB does not appears in Eq. (B.4),

because B is independent of the coordinate ri. The expressions of the gradients~∇riA, ~∇riP1 and ~∇riP2 with respect to the coordinate ri are given by

~∇riAA

=

NA∑j

~∇rifAA(|ri − rj|)fAA(|ri − rj|)

,

~∇riP1

P1

=

NB∑α=1

~∇rifAB(|ri − rα|)∑NB

α=1 fAB(|ri − rα|),

~∇riP2

P2

=

NB∑α=1

~∇rifAB(|ri − rα|)∑NA

i=1 fAB(|ri − rα|).

(B.5)

Now we are going to calculate the expressions for the Laplacians involved in

Eq. (B.1). The Laplacian of the trial wave function ∆riΨS(R) with respect to

the coordinate ri is given by

∆riΨS(R)

ΨS(R)=

[∆riAA

+ 2~∇riAA·~∇riP1 + ~∇riP2

P1 + P2

+∆riP1 + ∆riP2

P1 + P2

]. (B.6)

The second term in the right hand side of Eq. (B.6) is difficult to calculate in the

DMC code, since it involves the inner product of different quantities. To remove

119

this term let us to calculate the square of the drift force, from Eq. (B.4) we obtain

| ~Fri |2 =

(~∇riAA

)2

+ 2~∇riAA·~∇riP1 + ~∇riP2

P1 + P2

+

(~∇riP1 + ~∇riP2

P1 + P2

)2

. (B.7)

Now we solve for the inner product term

2~∇riAA·~∇riP1 + ~∇riP2

P1 + P2

= | ~Fri |2 −(~∇riAA

)2

−(~∇riP1 + ~∇riP2

P1 + P2

)2

. (B.8)

After substituting Eq. (B.8) into Eq. (B.6) we obtain a expression for the Lapla-

cian of the trial wave function ∆riΨS(R)

∆riΨS(R)

ΨS(R)=| ~Fri |2 +

∆riAA−(~∇riAA

)2

+∆riP1 + ∆riP2

P1 + P2

−(~∇riP1 + ~∇riP2

P1 + P2

)2

.

(B.9)

Here, we see that ∆riΨS(R) is given in terms of the gradients and Laplacians of

A, P1 and P2. We already derived expressions for the gradients, which are given

in Eq. (B.5). Now, we focus on obtaining expressions for ∆riA, ∆riP1 and ∆riP2.

The Laplacian ∆riA reads as

∆riAA

=

NA∑j

∆rifAA(|ri − rj|)fAA(|ri − rj|)

−(~∇rifAA(|ri − rj|)fAA(|ri − rj|)

)2

+

(NA∑j

~∇rifAA(|ri − rj|)fAA(|ri − rj|)

)2

.

(B.10)

The Laplacian ∆riP1 is given by the following expression

∆riP1

P1

=

NB∑α=1

∆rifAB(|ri − rα|)∑NB

α=1 fAB(|ri − rα|). (B.11)

120 Appendix B. Symmetric Trial Wave Function

And finally, the Laplacian ∆riP2 is given by

∆riP2

P2

=

NB∑α=1

∆rifAB(|ri − rα|)∑NA

i=1 fAB(|ri − rα|)−(


i=1 fAB(|ri − rα|)

)2

+

(NB∑α=1


i=1 fAB(|ri − rα|)

)2

.

(B.12)

The full Laplacian of the many-body trial wave function ∆riΨS(R) with respect

to the coordinate ri is given by

∆riΨS(R)

ΨS(R)=| ~Fri |2 +

NA∑j

∆rifAA(|ri − rj|)fAA(|ri − rj|)

−(~∇rifAA(|ri − rj|)fAA(|ri − rj|)

)2

+∆riP1 + ∆riP2

P1 + P2

−(~∇riP1 + ~∇riP2

P1 + P2

)2

.

(B.13)

Analogous to the Eq. (B.13), the full Laplacian of the many-body trial wave

function ∆rαΨS(R) with respect to the coordinate rα is given by

∆rαΨS(R)

ΨS(R)=|~Frα|2 +

NB∑β

∆rαfBB(|rα − rβ|)fBB(|rα − rβ|)

−(~∇rαfBB(|rα − rβ|)fBB(|rα − rβ|)

)2

+∆rαP1 + ∆rαP2

P1 + P2

−(~∇rαP1 + ~∇rαP2

P1 + P2

)2

.

(B.14)

Here, the drift force with respect to the coordinate rα is given by

~Frα =~∇rαΨ(R)

Ψ(R)=~∇rαBB

+~∇rαP1 + ~∇rαP2

P1 + P2

. (B.15)

In the following we are going to obtain the expressions for the gradients and

Laplacians involved in Eq. (B.14). The gradients ~∇rαB, ~∇rαP1 and ~∇rαP2 are

121

given by~∇rαBB

=

NB∑β

~∇rαfBB(|rα − rβ|)fBB(|rα − rβ|)

,

~∇rαP1

P1

=

NA∑i=1

~∇rαfAB(|ri − rα|)∑NB

α=1 fAB(|ri − rα|),

~∇rαP2

P2

=

NA∑i=1

~∇rαfAB(|ri − rα|)∑NA

i=1 fAB(|ri − rα|).

(B.16)

The Laplacian ∆rαB is given by

∆rαBB

=

NB∑β

∆rαfBB(|rα − rβ|)fBB(|rα − rβ|)

−(~∇rαfBB(|rα − rβ|)fBB(|rα − rβ|)

)2

+

(NB∑β

~∇rαfBB(|rα − rβ|)fBB(|rα − rβ|)

)2

.

(B.17)

The expression for the Laplacian ∆rαP1 reads as

∆rαP1

P1

=

NA∑i=1

∆rαfAB(|ri − rα|)∑NB

α=1 fAB(|ri − rα|)−(


α=1 fAB(|ri − rα|)

)2

+

(NA∑i=1


α=1 fAB(|ri − rα|)

)2

.

(B.18)

Finally, the Laplacian ∆rαP2 is given by

∆rαP2

P2

=

NA∑i=1

∆rαfAB(|ri − rα|)∑NA

i=1 fAB(|ri − rα|). (B.19)

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AGRADECIMIENTOS

Primero quiero agradecer a mis tutores, al Dr. Jordi Boronat y al Dr. Grigory

Astrakharchik, por darme la oportunidad de realizar este trabajo de investigacion

bajo su supervision y por permitirme formar parte del grupo de investigacion

Barcelona Quantum Monte Carlo. Ademas, agradezco a los miembros de este

grupo, los cuales ayudaron a mi formacion academica.

En especial agradezco a mis companeros y amigos: Huixia Lu, Juan Sanchez,

Raul Bombın, Viktor Cikojevic, y Giulia De Rosi, por hacer mas amenas las

comidas en la universidad. Tambien agradezco a mis amigos en Mexico, que a

pesar de los mas de 9000 Km de distancia nuestra amistad continua.

Agradezco al Consejo Nacional de Ciencia y Tecnologıa (CONACyT) y al

pueblo de Mexico por haberme otorgado una beca para realizar mis estudios de

doctorado en la Universitat Politecnica de Cataluna.

Dedico este trabajo a mi familia, por su carino y apoyo incondicional. Tambien

lo dedico a Esteban, por su apoyo en todo momento, y por los buenos momentos

que hemos pasado juntos en estos anos.

139

Documents

Quantum Monte Carlo study of few-and many-body bose